Continuity of a Function

Continuity of a Function

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  Gandhinagar Institute of Technology(012) Subject: Calculas (2110014) Active Learning Assignment Branch : Computer DIV. : A-2 Prepared by : - Vishvesh jasani (160120107042) Guided By: Prof. Nirav Pandya topic:continuity of a Function
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  Brief flow ofpresentation 1. Introduction 2. Intuitive Look at Continuity 3. Continuity at a Point 4. Continuity Theorem
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  Intuitive Look atContinuity A function without breaks or jumps The graph can be drawn without lifting the pencil 
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  Continuity at aPoint A function can be discontinuous at a point A hole in the function and the function not defined at that point A hole in the function, but the function is defined at that point
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  Continuity at aPoint A function can be discontinuous at a point The function jumps to a different value at a point The function goes to infinity at one or both sides of the point, known as a pole
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  Definition of Continuityat a Point A function is continuous at a point x = c if the following three conditions are met 1. f(c) is defined 2. 3. determine which of the conditions is violated in the examples of discontinuity lim ( ) exists lim ( ) ( ) x c x c f x f x f c    x = c
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  Continuity Theorem A functionwill be continuous at any number x = c for which f(c) is defined, when …  f(x) is a polynomial  f(x) is a power function  f(x) is a rational function  f(x) is a trigonometric function  f(x) is an inverse trigonometric function
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  Properties of ContinuousFunctions If f and g are functions, continuous at x = c Then …  is continuous (where s is a constant)  f(x) + g(x) is continuous  is continuous  is continuous  f(g(x)) is continuous ( )s f x ( ) ( )f x g x ( ) ( ) f x g x
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  One Sided Continuity Afunction is continuous from the right at a point x = a if and only if A function is continuous from the left at a point x = b if and only if lim ( ) ( ) x a f x f a   lim ( ) ( ) x b f x f b   a b
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  Continuity on anInterval The function f is said to be continuous on an open interval (a, b) if It is continuous at each number/point of the interval It is said to be continuous on a closed interval [a, b] if It is continuous at each number/point of the interval and It is continuous from the right at a and continuous from the left at b