# Continuity and Differentiability

## In calculus, continuity and differentiability of a function at a point are the determined by the limit of the function at that point. They are as given below: Continuous Function: A function ‘f ’ is said to be continuous at a point ‘a’ if and only if ‘f’ is defined in some neighborhood of a and limit of the function at the point a is equal to the value of the function at a. Let f and g be real valued continuous functions defined in a neighborhood of a point ‘a’. Then • kf is a continuous at ‘a’ for all real values of k. • f±g is a continuous at ‘a’ for all real values of k. • kg is a continuous at ‘a’ for all real values of k. • f/g is a continuous at ‘a’ for all real values of k, provided value of g at a is not equal to zero. The function which is not continuous at any point ‘c’, is said to be discontinuous at that point. The set of all points where the function is continuous, is called its domain of continuity. Let ‘f’ and ‘g’ be real functions such that ‘fog’ is defined at a. If ‘g’ is continuous at a point ‘a’ and if ‘f’ is continuous at g (a), then (fog) is continuous at ‘a’. • Rolle’s Theorem: If a function f :[a,b]®R such that a) continuous in the closed interval [a, b] b) differentiable in the open interval (a, b ) and c) f(a) = f(b) then there exists at least one real number c in (a, b) such that f´(c) = 0. • Lagrange’s Mean Value Theorem: If a function f :[a,b]®R such that a) continuous in the closed interval [a, b] and b) differentiable in the open interval (a, b) then there exists at least one real number c in (a, b) such that f´(c) = [f(b) – f(a)] /(b – a). Keywords: Differentiability

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