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icantadd
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differentiable and uniformly continuous??
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).
The definition of uniform continuity is:
for any e there is a d s.t. | x- Y | < d then | f(x) -f(y | < e.
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.
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.