Thread: Analysis Question

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

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

Originally Posted by Aryth

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

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