3a topics in differentiation - zerr abueg...differentiable. Note that nowhere differentiable functions are continuous. This is an “extreme case” of showing that the converse of

LC Abueg: mathematical economics

chapter 3a: topics in differentiattion 1

Chapter 3a

Topics in differentiation

Lectures in

Mathematical

Economics

L Cagandahan Abueg

De La Salle University

School of Economics

Problems in differentiation


( ) = − 2( ) 1i f x x

� Problem 1. Using the definition, find the derivative of the following functions:

( ) = 3( ) 5ii g x x

( ) −= 1( )iii h x x




→∞

+ =

1lim 1

n

ne

n

� Problem 2. Use L’Hospital’s Rule to give an alternate proof of the existence of Euler’s number:


( ) ( ) ( ), ,f a b f a f b a b+ = ∀ ∈R

Suppose that f(0) = 1 and f’(0) exists. Show that

′ ′= ∀ ∈( ) (0) ( ),f x f f x x R

If in addition, f’(0) = 1, what is f?

� Problem 10. Let f differentiable and real-valued with

The derivative

� Remark. It can be shown thati. if f(x) is differentiable, then

0

( ) ( )lim ( )

2h

f x h f x hf x

h→

+ − − ′=

ii. if f’(c) exists, then

( ) ( )lim ( ) ( )x c

xf c cf xf c cf c

x c→

− ′= −−



Trigonometric functions


� Theorem 3a.1. [Derivatives of trigonometric functions] For any x in the real line,

( ) sin cosd

i x xdx

=

( ) cos sind

ii x xdx

= −


( ) 2tan secd

iii x xdx

=

( ) 2cot cscd

iv x xdx

= −

( ) sec sec tand

v x x xdx

=

( ) csc csc cotd

vi x x xdx

= −




� Remark. For any x in the real line, observe that

( )4

4 sin sind

i x xdx

=

( )4

4 cos cosd

ii x xdx

=

Trig. & hyp. functions

� Exercise. Find the derivative of the following functions:

( ) ( ) cos(sin )i f x x=

( ) =( ) sin(exp )ii g x x

( ) =( ) tan(ln )iii h x x

One-sided derivatives




� Recall the concept of a one-

sided limit: from the epsilon-delta criterion for limit, it is possible that the direction of xin approaching a cluster point c may be only done in one direction.

� We review the corresponding criterion when x approaches cfrom the left [or from the right].

One-sided limits

� Definition. Let

lim ( )r

x cf x L

+→=

:A f A⊆ ∧ →R � R

If c is a cluster point of

iff ε δδε

∀ > ∃ >< − <

⇒ − <

0 0(0

( ) )

x c

f x L

{ }( , ) :A c x A x c∞ = ∈ > ∩

then

Lr is the right-hand limit of f at c.

One-sided limits

� Definition. Let

lim ( )l

x cf x L

−→=

:A f A⊆ ∧ →R � R

If c is a cluster point of

iff ε δδε

∀ > ∃ >< − <

⇒ − <

0 0(0

( ) )

c x

f x L

{ }( , ) :A c x A x c−∞ = ∈ < ∩

then

Ll is the left-hand limit of f at c.



One-sided limits

L

cx c+→x c−→


� Definition. A function f is said to be differentiable from the left of x if

0

( ) ( )limh

f x h f x

h−→

+ −

exists. We call this the left derivative

of f at x, and we denote

0

( ) ( )( ) lim

h

f x h f xf x

h−−→

+ −′ =


� Definition. A function f is said to be differentiable from the right of x if

0

( ) ( )limh

f x h f x

h+→

+ −

exists. We call this the right

derivative of f at x, and we denote

0

( ) ( )( ) lim

h

f x h f xf x

h++→

+ −′ =



One-sided limits

� Theorem 3a.3. We say that a function f is differentiable at x iff

( ) ( ) ( )f x f x f x+ −′ ′ ′= =

i.e., the right and left derivatives coincide.

Piecewise functions

Piecewise functions

� In mathematical literature, given a differentiable function

:f I → Ron a closed and bounded interval I = [a,b] with

a b−∞ < < < ∞it is usual [and logical] that f is differentiable on (a,b) and continuous on [a,b].



Piecewise functions

Recall that

( ) ,f x x x= ∀ ∈R

at the point x = 0, f is not differentiable since by the uniqueness of limits (implying a unique tangent line at a point on f), this unique tangent line does not exist. We call the point x = 0 in this case a sharp point.

Piecewise functions

However, note that on the subintervals

( ,0), (0, )−∞ ∞the function does not have sharp points. Moreover, the absolute value function is differentiable on each of the subintervals above.

Piecewise functions

x

y

= ∈( ) ,f x x x R



Piecewise functions

� Definition. If a function : [ , ]f a b → Rhas a finite number of points of discontinuity given by x1,x2,…,xk, such that f is continuous on each subinterval

+1 1( , ),...,( , ),...,( , )i i k

a x x x x b

for i = 1,2,…,k – 1 and possibly defined on some (or all) points of discontinuity, we say that f is a piecewise continuous function.

Piecewise functions

� Definition. Consider a function

→: [ , ]f a b R

If

: ( , )f a b′ → Rexists, we say that f is a smooth

function. If there is a finite number of points given by x1,x2,…,xk, such that f’ exists on each subinterval

Piecewise functions

1 1( , ),...,( , ),...,( , )i i k

a x x x x b+

for i = 1,2,…,k – 1, we say that f is a piecewise smooth function.

� Remark. If a function is piecewise smooth, it is also piecewise continuous (immediate from Theorem 3.1). The converse does not hold.



Piecewise functions

� Example. Consider the function

1( ) , *f x x

x= ∈R

[i.e., f is defined on the nonzero reals]. Note that

2

1( ) , *f x x

x′ = − ∈R

Piecewise functions

x

y

−→ 0x

+→ 0x

−→= −∞

0lim ( )x

f x

+→= +∞

0lim ( )x

f x

1( ) , *f x x

x= ∈R

Piecewise functions

x

y

−→ 0x +→ 0x

0lim ( )x

f x−→

′ = −∞0

lim ( )x

f x+→

′ = −∞

2

1( ) , *f x x

x′ = − ∈R



Piecewise functions

� Definition. Consider a piecewise smooth function f and a finite number of sharp points [or points of discontinuity]

1 2, ,...,k

x x x

If f is linear in each of the subintervals

1 1( , ),...,( , ),...,( , )i i k

a x x x x b+

Piecewise functions

[possibly defined on some or all of the sharp points or points of discontinuity]. We then call f a piecewise linear function.

Piecewise functions

� Example. The absolute value function

( ) ,f x x x= ∈R

is a piecewise linear function. Note that this not only piecewise continuous, but is continuous in the whole of the real line.



Piecewise functions


2( ) 1,f x x x= − ∈R

By definition,

− ∈ −∞ − ∞= − ∈ −

∪2

2

1 ( , 1] [1, )( )

1 ( 1,1)x x

f xx x

Piecewise functions

x

y

(1,1)

(–1,0) (1,0)

2( ) 1f x x= −

Piecewise functions

Note that f is not differentiable at the points

1, 1x = −i.e., f is piecewise smooth. The derivatives of f on the particular intervals are,

2 ( , 1) (1, )( )

2 ( 1,1)x x

f xx x

∈ −∞ − ∞′ = − ∈ −

∪



Piecewise functions

x

y

(–1, 2)

(–2, –4)

(1, –2)(–1, –2)

(1, 2)

(2, 4)

(0,0)

Since f’ is not differentiable at the points x = 1, –1, clearly the graph of f is not continuous at these points in the domain of f.

Piecewise functions

� Example. Consider a piecewise linear function given by

1( ) [ , ]

{1,2,..., 1}m m m m

f x a b x x x x

m k

−= + ∈∈ +

with k a fixed positive integer and

0 1,k

x a x b+= =

Piecewise functions

Obtaining the derivative of f respective to subintervals, we have

1( ) [ , ]

{1,2,..., 1}m m m

f x b x x x

m k

−′ = ∈∈ +

Let ( ) ( )g x f x′=

The function g is a particular example of a class of functions important in real analysis.



Piecewise functions

� Definition. A function defined by

1

1

( , )( )

{1,2,..., }

m m m

m m

b x x xh x

c x x

m k

−

−

∈= =

∈

where k a fixed positive integer is called a step function.

Piecewise functions

x

y

x1 x2 x5x3 x4x0

b1

b2

b3

b4

b5c0

c1

c2

c3

c4

c5

a step function

Piecewise functions

� Example. The function given by

= � �� ( )k x x

where

, ( 1, ],x n x n n n= ∈ − ∀ ∈� �� Z

is called the greatest integer

function.



Piecewise functions

x

y

1 2 53 40

1

( )k x x= � ��

2

3

4

5

A circle with shade indicates that only the rightmost endpoints are included in each “step”.

Piecewise functions

� Example. Consider again the absolute value function

( ) ,f x x x= ∈R

Since f above is piecewise smooth. By obtaining the derivative of f on respective subintervals, we get

Piecewise functions

1 0( )

1 0x

f xx

>′ = − <

By “defining” f’(x) = 0 [and naming this function g, we have

1 0( ) 0 0

1 0

x

g x x

x

>= =− <



Piecewise functions

1

-1

0

x

y

Observe that this function g is precisely the signum function

denoted sgn(x), which is also a step function.

Piecewise functions

� Example. [Weierstrass] There is a function f continuous on the real line but is not differentiable at any point in its domain. This function is given by

0

1( ) cos( )n

nn

f x b xa

π∞

=

=∑

� Proof. See Wade [2010], pp. 264-266. █

Piecewise functions

� Remark. The function provided by Weierstrass in the previous example is a case of a class of functions called nowhere

differentiable. Note that nowhere differentiable functions are continuous. This is an “extreme case” of showing that the converse of Theorem 3.1 is not true.



Uniform derivatives

Uniform derivatives

� Remark. Recall the definition of uniform continuity from §2a:

Let

0 0

( , 0 ( ) ( ) )x c A x c f x f cε

ε δδ ε

∀ > ∃ >∈ ∧ < − < ⇒ − <

We say that f is uniformly continuous on A iff

:A f A⊆ ∧ →R � R

Analogously, we can define the concept of uniform continuity given in the next slide:

Uniform derivatives

� Definition. A function

:f I → Ron a closed and bounded interval I is said to be uniformly

differentiable on I iff for every ε > 0, there is a δ > 0 such that if 0 < |x – y| < δ, then

( ) ( )( )

f x f yf x

x yε− ′− <

−



Uniform derivatives

� Remark. From the definition of uniform differentiability, it requires that the function f is differentiable, as seen in the expression

( ) ( )( )

f x f yf x

x yε− ′− <

−

Thus, if a function is uniformly differentiable, it is also differentiable.

Uniform derivatives

� Exercise. Prove or give a counterexample: the differentiability of a function f is sufficient for f to be uniformly differentiable.

Uniform derivatives

� Theorem 3a.4. If

:f I → Ris uniformly differentiable on I, then f’ is continuous on I.

� Proof. Obvious [immediately following from the previous remark, and Theorem 3.1]. █



Uniform derivatives

Show that if f is differentiable on Iand f’ is bounded on I, then fsatisfies the Lipschitz condition.

:f I → R

� Exercise. Let

Carathédory’s theorem

Carathédory's theorem

� Remark. Recall the mean value theorem (Theorem 3.19): Let f be a continuous function on [a,b], differentiable on (a,b), and let a < b. Then there exists a real number c in (a,b) such that

−′ =−

( ) ( )( )

f b f af c

b a




Reexpressing the mean value theorem, we have

( ) ( ) ( )f c b a f b f a′ − = −

Note that the existence of f’(c) is not always guaranteed to exist for any function f. The necessary conditions are stated in the next theorem, due to Carathédory.


� Theorem 3a.5. [Carathédory] Let fbe a function defined on I = [a,b] and let c be a point in [a,b]. Then f is differentiable at c iff there is a function

: [ , ]a bϕ → Rthat is continuous at c and satisfies

( ) ( ) ( ),x x c f x f c x Iϕ − = − ∀ ∈


� Example. Consider the function3( )f x x=

Note that3( )f c c=

From elementary algebra,

3 3 2 2( )( )x c x c x cx c− = − + +




Using Carathédory’s theorem

3 3( ) ( )f x f c x c− = −2 2( )( )x c x cx c= − + +

( ) ( )x c cϕ= −

Thus,

2 2( ) ( )x x cx cϕ = + +


Observe that

2 2( ) 3 ( ) 3f x x f c c′ ′= ⇒ =and

2 2( )x x cx cϕ = + +2 2( ) ( )c c c c cϕ⇒ = + +2 2 2c c c= + +

23c= ( )f c′=


which is an expected result since

3( )f x x=is a continuously differentiable function for any real number c.




� Example. Using a similar argument, the function

2( )g x x=satisfies Carathédory’s theorem with

( )x x cϕ = +

Directional derivatives

Directional derivative

� We first recall the definition of a partial derivative in §3 and the corresponding geometric interpretation, given in the next slides.



Partial derivative

� Definition. Let f be a function of several variables, say x1,x2,..., xn. The partial derivative of f at xj is given as

1 1

0

( ,..., ,..., ) ( ,..., )lim j n n

hj

f x x h x f x xy

x h→

+ −∂ =∂

if this limit exists. If all partial derivatives with respect to the nvariables exist, then the n-tuple

Partial derivative

1 2

, ,...,n

y y y

x x x

∂ ∂ ∂ ∂ ∂ ∂

is called the gradient of f at the point (x1,x2,...,xn). This is also denoted as

f∇read as “del f”.

Partial derivative

� Remark. Given f be a function of several variables, say x1,x2, ..., xn, we denote the corresponding partial derivative with respect to xj as

jx

j

yf

x

∂ ′=∂



Geometric interpretation

x

y

z

f(x,y)

l

t

A

B

C

D

fx‘(x,y) is the slope of the curve BDC on the surface above the line l parallel to the x-axis;


� Remark. The idea that the partial derivative

jx

j

yf

x

∂ ′=∂

is the rate of change of f in the direction of the xj-axis may be generalized to any direction represented by a vector d ≠ θ.


i.e.,

1 2( , ,..., ) (0,0,...,0)n n

d d d= ≠ =d θ

Without loss of generality, we may consider a vector d whose length

is 1, i.e.,

2

1

1n

kk

d=

= = ∑d




We wish to determine the rate of change f in the direction d. To this end, we formally define the derivative of f at x = (x1,x2,…,xn) in the direction d by considering the values of fon the line x + pd, i.e., f(x + pd) for small values of p.


� Definition. Let f be a real valued function of n variables, i.e.,

: nf →R R

and let d be a vector such that its length is 1. The derivative of f at xin the direction d, defined by

0

( ) ( )( ) lim

p

f p ff

p→

+ −′ =d

x d xx


if this limit exists. We call this derivative a directional derivative.

1

( )n

kk k

ff d

x=

∂′ =∂∑d

x

� Theorem 3a.6. If f is differentiable at x, then




1

( )n

kk k

ff d

x=

∂′ =∂∑d

x

� Remark. In the language of linear algebra,

T[ ( )]f= ∇ x d [ ( )]f= ∇ •x d

i.e., a directional derivative is a dot product of the gradient of fand the vector d.


(0,0,...,0,1,0,...,0)k= =d u

� Corollary 3a.7. If

i.e., d is the unit vector (such that the kth entry is 1 and the rest are zeroes), then

[ ( )] , 1,2,...,k

k

ff k n

x

∂∇ • = =∂

x u


( )n

f∇ ≠x θ


then the gradient of f is the direction of maximum rate of increase of f.




[ ( )] 0f∇ • >x d

� Corollary 3a.9. Let d be a vector such that its length is 1 and

Then there exists a q > 0 such that

( ) ( ), 0f p f p q+ > < <x d x


[ ( )] 0f∇ • >x d

� Remark. The above corollary says that if

(i.e., d has the same general direction as the gradient of f in the sense that the angle θ formed by these two vectors is less than π/2 radians), then any small movement in the direction d will


( )n

f∇ ≠x θ

increase f. Thus, if the gradient

then this points in the direction of increasing values of f(i.e., the gradient points uphill).




( , )z f x y=� Definition. Consider a relation

given a fixed constant z. Let [a,b] be a subset of the graph of f. The set

{ }: ( , ), [ , ]C y z f x y x a b= = ∈

is called a curve in the Cartesian plane.


( )( ) ( ), ( )f t x t y t=We call

a parametric representation of f. If x and y are differentiable functions of t, we call the curve C a differentiable curve.


� Remark. Consider differentiable curve C in the Cartesian plane. The function

( , )z f x y=may be represented implicitly by an equation

0 ( , )g x y=




At a point (x,y), the tangent to the curve has slope

dy dy dt

dx dx dt= ⋅ /

/dy dt

dx dt= ( )

( )y t

x t

′=

′

and observe that

( )( ) ( ), ( )f t x t y t′ ′ ′=


which is just a ray containing the origin (0,0) and the point (x,y). Thus, the tangent to the curve Cand f’(t) have the same slopes and if we translate f’(t) to (x,y), then f’(t) will be tangent to C. Thus, we call f’(t) a tangent

vector to the curve C.


By totally differentiating

0 ( , )g x y=

(i.e., applying chain rule with respect to t), we have

0g dx g dy

x dt y dt

∂ ∂= ⋅ + ⋅∂ ∂




In the language of linear algebra,

0g dx g dy

x dt y dt

∂ ∂= ⋅ + ⋅∂ ∂

dx

g g dt

x y dy

dt

∂ ∂ = ∂ ∂

[ ] ( )g f t′= ∇ •


i.e., the gradient of g and the tangent vector f’ are orthogonal, at the point (x,y). It can be shown that this is true if

nC ⊆ R

Homothetic functions




� Recall the definition of a homogeneous function f:

A function f is said to be homogeneous of degree k iff

( ) ( ), 0kf tx t f x t= ∀ >

If k = 1, we then say that the function f is linearly homogeneous.


� From a remark in implicit function theorems, if

is a homogeneous function of degree k, then the slopes of the level curves are the same at each point on the ray from the origin.

( , )z f x y=


x

y

0 0 0( , )z f x y=(0,0)

(tx0,ty0)

y0

x0

0 0

0 0

( , )

( , )x

y

f x y

f x y

′−

′

(x0,y0)

tx0

ty0

0 0( , )t

z f tx ty=




� This property is possessed by a larger class of functions that contain the homogeneous functions as a subclass. These are called homothetic functions, which we formally define in the next slide.

� Before stating this definition, recall the class of monotone functions in §2.1.

Monotone functions

� Definition. A function f is said to be weakly monotone iff f is either nondecreasing or nonincreasing. A function f is said to be monotone (or strictly

monotone) iff f is either increasing or decreasing.

Monotone functions

a nondecreasing function

The “flat” portions of the graph of f makes it nondecreasing



Monotone functions

a nonincreasing function

The “flat” portions of the graph of f makes it nonincreasing



: , nf D D→ ⊆R R

is said to be homothetic iff there is a monotone (increasing) function

:h →R R

and a homogeneous function

:g D → R


such that

( ) ( ) ( )f h g g= = x x h x�

where

1 2( , ,..., )n

x x x=x




� Remark. In the definition of a homothetic function f (and the corresponding homogeneous function g), if

: , ng D D→ ⊆R R

then D satisfies the condition

{ }0D t t D∈ ∧ > ⇒ ∈x x


This translates to the usual problem of production functions (that are homogenous of some degree) in microeconomics: the problem of feasibility.



2:f ++ →R R

given by

( )3 31 2( ) ln 4f x x= +x

Since

3 31 2( ) 4g x x= +x




is homogeneous of degree 3 and

( ) ln( )h z z=is monotone (i.e., increasing), then f is homothetic. But observe that f is not homogeneous:

3( ) 3ln ( ) ( )f t t f t f= + ≠x x x


� Theorem 3a.10. Every homogeneous function is homothetic.

� Remark. The above theorem stresses the fact that the class of homogeneous functions is a subset of the class of homothetic functions.


� Theorem 3a.11. A nonnegative-valued function f and homogeneous of degree k > 0 can be expressed as a homothetic function

f h g= �

where g is linearly homogeneous.




� Remark. Production functions belong to the class of functions that are homogeneous of positive degree. From §3, we have noted that the degree of homogeneity of a production is its returns to scale; thus, it makes economic sense if this degree is positive.


Thus, by the previous theorem, any production function can be expressed as a composition of two functions h and g, where the function g is linearly

homogeneous.


� Theorem 3a.12. The slopes of level curves of a homothetic function are the same at every point of a ray emanating from the origin.




� Remark. In consumer theory, given a utility function U(x) and ain increasing function g(x), then the composition h given by

( )( ): ( )h x g U= x

preserves ordering of preferences (i.e., ordinal utility).


Moreover, if in addition U is homogeneous of degree k [with respect to the consumption bundle x], then

( )( ): ( )h x g U= x

is a homothetic function. IN microeconomics, we call h a monotonic transformation of U.


� Theorem 3a.13. If f is a homothetic function with

1 2, nD≠ ∈ ⊆x x R

then

1 2( ) ( )f f=x x

implies

1 2( ) ( ), 0f fα α α= ∀ >x x




� Remark. In the previous theorem, if f is a production function, then if

1 2, nD≠ ∈ ⊆x x R

are two input vectors that produce the same output, i.e.,

1 2( ) ( )q f f= =x x


then

1 2( ) ( ), 0q f fα α α= = ∀ >x x

i.e.,

1 2,α αx x

will produce the same output for every positive α.

Functions from Cn[a,b]




� Recall that a differentiable function y = f(x) may be again differentiable [under some (necessary) conditions], which may give rise to a new differentiable function.

� This leads us to the most commonly used terminologies for characterizing functions in economics.


� Recall that in §3, we have the following definition:

If f is a differentiable function and f’ exists, we say that f is differentiable. If f’is a continuous function, we say that f iscontinuously differentiable. If f’ is again differentiable, and f” exists, we say thatf is twice differentiable. If f” is a continuous function, we say that f is twice continuously differentiable.


� It is always of interest in economics [which is also desired] that functions depicting economic behavior are twice continuously differentiable.

� We formally define a phrase that characterizes economic behavior pertaining to a particular variable of interest given a function.




� Definition. We say that a function y = f(x) is well-behaved whenever

( ) ( ) 0, domi f x x f′ ≥ ∀ ∈

( ) ( ) 0, domii f x x f′′ < ∀ ∈


� Definition. We say that a function z = g(x1,x2,…,xn) is well-behaved

[with respect to x0] whenever

( ) 0( )i f∇ ≥x θ

( ) 2 0 0( ) ( )

is negative definitef

ii f H∇ =x x

Functions from Cn[a,b] 2 2 2

22 1 11

2 2 2

21 2 22

2 2 2

21 2

( )

n

g n

n n n

f f f

x x x xx

f f f

H x x x xx

f f f

x x x x x

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂

= ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

x

⋯

⋯

⋮ ⋮ ⋱ ⋮

⋯





( )y f x=

is said to be nth-differentiable iff

( )( )n

n

n

d yf x

dx=

exists. If g is a function having the following properties:


( )

( )

exists on ( , )

continuous on [ , ]

n

n

n

n

d gi a b

dx

d gii a b

dx

then we say g belongs to the class

of nth-continuously differentiable

functions on [a,b], i.e.,

[ , ]ng C a b∈


We also have the following special classes of functions:

i. If n = 0, then C0[a,b] is the class of continuous

functions on [a,b].ii. If n = 1, then C1[a,b] is the

class of continuously differentiable functions on

[a,b].




iii. If n = 2, then C2[a,b] is the class of twice continuously

differentiable functions on [a,b].

� Remark. Observe that most of the properties of differentiable functions in §3 follow the desired properties of functions in Cn[a,b]: differentiable in (a,b) and continuous in [a,b].


� Theorem 3a.14. If

( ) [ , ]ny f x C a b= ∈then f has a Taylor approximation of the form

( )0

0 01

( )( ) ( ) ( )

!

jnj

j

f xf x f x x x

j=

≈ + −∑

0 domx f∈



( ) [ , ]ny f x C a b= ∈and f and its derivatives up to the nth order are defined at zero, then f has the Maclaurinapproximation

( )

1

(0)( ) (0)

!

jnj

j

ff x f x

j=

≈ +∑




−

=

=

∑( ) ( ) ( )

0

( ) ( ) ( ) ( )n

n n k k

k

nfg x f x g x

k

� Exercise. Prove the Leibniz’ Rule for the nth derivative of a product:

[Real] analytic functions

Analytic functions

� Definition. A real-valued function f is said to be [real] analytic on a nonempty, open interval (a,b) iffgiven x0 ∈ (a,b), there is a power series centered at x0 which converges to f near x0 ; i.e., iffthere exist coefficients

{ }k ka

∈N



Analytic functions

and points c, d ∈ (a,b), such that c < x0 < d and

00

( ) ( ) , ( , )k

kk

f x a x x x c d∞

=

= − ∀ ∈∑

� Remark. We apply some “conventions” from the previous definition: that f(0) = f, and from factorials, 0! = 1.

Analytic functions

� Exercise. Let f(x) be an infinitely differentiable function such that for all reals, we have f’(x) = f(x) and that f(0) = 1. What is f?

Analytic functions

� Theorem 3a.16. Let c, d be extended real numbers with c < d, let x0 ∈ (c,d), and suppose that

: ( , )f c d → R

If

00

( ) ( ) , ( , )k

kk

f x a x x x c d∞

=

= − ∀ ∈∑



Analytic functions

then

( , )f C c d∞∈and

( )0( )

, {0}!

k

k

f xa k

k= ∀ ∈ ∪N

Analytic functions

� Remark. The previous theorem yields the Taylor expansion [or Taylor series representation] of f centered at x0. Note also that this implies that

( , )f C a b∞∈This is in contrast to the version in §3: differentiation of f is only up to the (n + 1)-order.

Analytic functions

� Definition. Let f ∈ C∞(a,b) . The Taylor series expansion of f about

x0 ∈ (a,b) is the infinite sum

( )0

01

( )( )

!

kk

k

f xx x

k

∞

=

−∑

If x0 = 0, the above infinite sum is called a Maclaurin series

expansion of f.



Analytic functions

� Remark. [A.L. Cauchy] The function

2exp( ) 0( )

0 0x x

f xx

− − ≠= =

has no Taylor approximation around the point x = 0.

Analytic functions

� Definition. Let f ∈ C∞(a,b) and x0 ∈ (a,b). The remainder term of

order n of the Taylor expansion of f centered at x0 is the function

0,

( )10

01

( ) ( )

( ): ( ) ( )

!

f x

n n

knk

k

R x R x

f xf x x x

k

−

=

=

= − −∑

Analytic functions

� Remark. From Theorem 3a.16 and the definition of the remainder, a function f ∈ C∞(a,b) is analytic on (a,b) iff for each x0 ∈ (a,b) there is an interval (c,d) containing x0such that

0,lim ( ) 0, ( , )f x

nn

R x x c d→∞

= ∀ ∈



Analytic functions

Recall that from Theorem 3.17, for every pair of points x, x0 in (a,b), there is a p between x, x0such that

( 1)1

1 0

( )( )

( 1)!

nn

n

f pR x x

n

++

+ = −+

where n is shifted to n + 1.

Analytic functions

� Theorem 3a.17. Let f ∈ C∞(a,b). If there is an M > 0 such that

( )( ) , ( , ),n nf x M x a b n≤ ∀ ∈ ∀ ∈N

then f is analytic on (a,b). In fact, for each x0 ∈ (a,b),

00

( ) ( ) , ( , )k

kk

f x a x x x a b∞

=

= − ∀ ∈∑

Analytic functions

� Example. The sine, cosine, and natural exponential functions are analytic on the whole of the real line and have the following Maclaurin series representations:

( )0

exp!

k

k

xi x

k

∞

=

=∑



Analytic functions

( )2

0

( 1)cos

(2 )!

k k

k

xii x

k

∞

=

−=∑

( )2 1

0

( 1)sin

(2 1)!

k k

k

xiii x

k

+∞

=

−=+∑

Analytic functions

� Theorem 3a.18. Let I be an open interval and let c be the center of this interval. Suppose further that

0

( ) ( ) ,k

kk

f x a x c x I∞

=

= − ∀ ∈∑

If x0 ∈ I and r > 0 satisfy

0 0( , )x r x r I− + ⊆

Analytic functions

then

( )0

00

0 0

( )( ) ( ) ,

!( , )

kk

k

f xf x x x

k

x x r x r

∞

=

= −

∀ ∈ − +

∑

In particular, if f ∈ C∞ whose Taylor expansion converges to f on some open interval J, then f is analytic on J.



Analytic functions

� Example. The natural logarithmic function has Taylor series representation

1

1

( 1)ln ( 1) , (0,2)

kk

k

x x xk

+∞

=

−= − ∈∑

Analytic functions

� Example. The function ax is analytic on the whole real line and has a Taylor series representation

1

(ln ), 0

!

kx k

k

aa x a

k

∞

=

= >∑

Analytic functions

� Theorem 3a.19. [Bernstein] If f ∈ C∞(a,b) and f(n)(x) > 0 for all x ∈ (a,b) and for all natural numbers n, then f is analytic on (a,b). In fact, if x0 ∈ (a,b) and f(n)(x) > 0 for all x ∈ [x0,b) and for all natural numbers n,

( )0

0 01

( )( ) ( ) , [ , )

!

kk

k

f xf x x x x x b

k

∞

=

= − ∀ ∈∑



Analytic functions

� Lemma 3a.20. Suppose that f and g are analytic on the open interval (c,d) and let x0 ∈ (c,d). If

0( ) ( ), ( , )f x g x x c x= ∀ ∈

then there is a δ > 0 such that

0 0( ) ( ), ( , )f x g x x x xδ δ= ∀ ∈ − +

Analytic functions

� Theorem 3a.21. [Analytic continuation] Suppose that I and Jare open intervals, that f is analytic on I, that g is analytic on J, and that a, < b are points in the intersection of I and J. If

( ) ( ), ( , )f x g x x a b= ∀ ∈then

( ) ( ),f x g x x I J= ∀ ∈ ∩

To end...

Pure mathematics is on the whole

distinctively more useful than applied. For what is useful above all is technique,

and mathematical technique is taught

mainly through pure mathematics.

GH Hardy [1877-1947]

Documents

3a topics in differentiation - zerr abueg...differentiable. Note that nowhere differentiable functions are continuous. This is an “extreme case” of showing that the converse of