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Limits and Continuity – Intutive Approach– Chapter 8 Paper 4: Quantitative Aptitude- Mathematics
Ms. Ritu Gupta B.A. (Hons.) Maths and MA (Maths)
Continuity
• Fundamental Knowledge • Its application
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Concept of Continuity
A function is said to be continuous at a point x = a if
(i) f(x) is defined at x = a
(ii)
(iii)
i.e. L.H.L = R.H.L = value of the function at x = a
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Discontinuous Function
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Properties of Continuous Functions
(i) If f(x) and g(x) are both continuous at a point x=a, then f(x)+g(x) is also continuous.
(ii) If f(x) and g(x) are both continuous at a point x=a then f(x) – g(x) is also continuous.
(iii) If f(x) and g(x) are both continuous at a point x=a then their product f(x) . g(x) is also continuous at x = a.
(iv) If f(x) and g(x) are both continuous at a point x=a then their quotient f(x)/ g(x) is also continuous at x=a and g(a) ≠ 0.
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Properties of Continuous Functions-Continued
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Illustrations
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Illustration 24
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Illustration 24 - Continued
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Illustration 24 - Continued
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Illustration 25
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Illustration 25 – Continued
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Illustration 26
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Illustration 26 – Continued
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Illustration 27
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Illustration 27 – Continued
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Illustration 28
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Illustration 28 – Continued
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Illustration 29
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Illustration 29 – Continued
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Illustration 29 – Continued
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Illustration 29 – Continued
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Illustration 30 A function f(x) is defined as follows:- f(x) = x2 when 0<x<1 = x when 1≤x<2 = (¼) x3 when 2≤x<3 Now f(x) is continuous at (a) x = 1 (b) x = 3 (c) x=0 (d) None of these Solution: Let us first check the continuity of a function f(x), at x = 1
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Illustration 30 - Continued
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Illustration 31
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Illustration 31 - Continued
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Illustration 31 - Continued
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Illustration 32
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Illustration 32 – Continued
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Illustration 33
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Illustration 33 – Continued
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Illustration 34
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Illustration 34 – Continued
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Illustration 34 – Continued
Illustration 35
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Illustration 35 – Continued
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Illustration 35 – Continued
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Illustration 36
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Illustration 36 – Continued
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Illustration 36 – Continued
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Illustration 37
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Illustration 37 – Continued
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Thank you
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