Thread: Proof Uniformly Continous

Need help with this proof. Thanks so much!

Let j be an interval and f a differentiable function on j whose derivative is bounded. Show that f is uniformly continuous (use the mean value theorem).
Observe that boundedness of f ' is not necessary for uniform continuity of f. Indeed, the function x ↦ sqrtx (x > 0) is uniformly continuous and differentiable, but its derivative is not bounded.

Originally Posted by ml692787

Need help with this proof. Thanks so much!

Let j be an interval and f a differentiable function on j whose derivative is bounded. Show that f is uniformly continuous (use the mean value theorem).

Let be an open inverval. Say is a differenciable function. And is bounded by positive constant. Let be two distinct points. Then so if we choose where this will complete the proof.

Originally Posted by ml692787

Observe that boundedness of f ' is not necessary for uniform continuity of f. Indeed, the function x ↦ sqrtx (x > 0) is uniformly continuous and differentiable, but its derivative is not bounded.

For any choose then say because otherwise this difference is zero which is certainly less than . In that case we get .