Analysis Question

**Aryth** · Nov 16th 2011, 11:03 AM

Suppose $f : (a,b) \to R$ is differentiable and $| f'(x)|\leq M, \forall x \in (a,b)$ . Prove that f is uniformly continuous on (a,b).

I am having trouble finding a starting point. Any help would be appreciated.

**Plato** · Nov 16th 2011, 11:10 AM

Originally Posted by Aryth

Suppose $f : (a,b) \to R$ is differentiable and $| f'(x)|\leq M, \forall x \in (a,b)$ . Prove that f is uniformly continuous on (a,b).

Use the mean value theorem.
If $(x,y)\subseteq (a,b)$ then $\left( {\exists z \in (x,y)} \right)\left[ {\left| {\frac{{f(y) - f(x)}}{{y - x}}} \right| = \left| {f'(z)} \right|} \right]$

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