Question

3)Let \(f\) differentiable on the open interval \(I\) and let \(f^{'}\)be bounded on \(I\).
b) Show that \(f\) is uniformly continuous on \(I\)

Answer 1

By the previous problem, we have
|f(x)-f(y) | < M |x-y|
Take \( \epsilon > 0\) and take \( \delta =\frac \epsilon M \) if
\( |x-y| < \delta \) then \( |f(x) - f(y)| < \epsilon\)

Answer 2

thank you very much Mr Elias,