Differentiable and uniformly continuous?

[icantadd]

differentiable and uniformly continuous??

Homework Statement

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).

Homework 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.

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.

 

[dirk_mec1]

Homework Statement

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).

Homework 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.

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.

Watch this:
<br /> |f(x) -f(y)| = \frac{|f(x) -f(y)|}{|x -y|} \cdot |x-y| &lt; M \cdot \delta <br />

Can you verify these steps?

 

[icantadd]

Watch this:
<br /> |f(x) -f(y)| = \frac{|f(x) -f(y)|}{|x -y|} \cdot |x-y| &lt; M \cdot \delta <br />

Can you verify these steps?

So long | x-y | not equal 0.

Yeah, I got the same thing. Actually a little bit differently, I used the mean value theorem, because you don't know that the function is bounded, only its derivative. The mean value theorem gets you to the point where you know that f '(c) = [f(x) - f(y)] / (x-y) is <= M (because f '(x) <= M for all x). Then you just multiply both sides by | x - y | and get the same end result. As long as | x - y| < e/M |f(x) - f(y) | < e.

Thank you for your help!