A differentiable function f: I→R is said to be uniformly differentiable on I=[a, b] if for every ɛ>0 there exists δ>0 such that if 0< |x-y|<δ and x, y ∊ I then |(f(x)-f(y))/(x-y) - f'(x)| < ɛ. Show that if f is uniformly differentiable on I, then f' is continuous on I.

A differentiable function f: I→R is said to be uniformly differentiable on I=[a, b] if for every ɛ>0 there exists δ>0 such that if 0< |x-y|<δ and x, y ∊ I then |(f(x)-f(y))/(x-y) - f'(x)| < ɛ. Show that if f is uniformly differentiable on I, then f' is continuous on I.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter5: Graphs And The Derivative

Section5.3: Higher Derivatives, Concavity, And The Second Derivative Test

Problem 61E

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With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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