- #1

Aryth

- 8

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## Homework Statement

A function [itex]f:(a,b)\to R[/itex] is said to be uniformly differentiable iff [itex]f[/itex] is differentiable on [itex](a,b)[/itex] and for each [itex]\epsilon > 0[/itex], there is a [itex]\delta > 0[/itex] such that [itex]0 < |x - y| < \delta[/itex] and [itex]x,y \in (a,b)[/itex] imply that [itex]\left|\frac{f(x) - f(y)}{x - y}-f'(x)\right| < \epsilon[/itex].

Prove that if f is uniformly differentiable on [itex](a,b)[/itex], then [itex]f'[/itex] is continuous on [itex](a,b)[/itex].

## The Attempt at a Solution

This is my first time being presented with the definition of uniform differentiability. I suppose that I am looking to show that the definition of uniform differentiability implies [itex]|f'(y) - f'(x)|< \epsilon[/itex]... However, I'm having a hard time doing that. Any help would be appreciated.