# Differentiable and uniformly continuous?

1. Nov 10, 2008

differentiable and uniformly continuous??

1. The problem statement, all variables and given/known data
Suppose f:(a,b) -> R is differentiable and | f'(x) | <= M for all x in (a,b). Prove f is uniformly continuous on (a,b).

2. Relevant equations
The definition of uniform continuity is:
for any e there is a d s.t. | x- Y | < d then | f(x) -f(y | < e.

3. The attempt at a solution
Intuitively, if f is differentiable it is continuous. If its derivative is bounded it cannot change fast enough to break continuity. The interval is bounded, and the function must be bounded on the open interval. It seems that there is not way that the function cannot be uniformly continuous. But how do I say that? Or am I on the wrong track altogether.

2. Nov 10, 2008

### dirk_mec1

Re: differentiable and uniformly continuous??

Watch this:
$$|f(x) -f(y)| = \frac{|f(x) -f(y)|}{|x -y|} \cdot |x-y| < M \cdot \delta$$

Can you verify these steps?

3. Nov 10, 2008