Question

3)Let \(f\) differentiable on the open interval \(I\) and let \(f^{'}\)be bounded on \(I\).
a) Show that \(|f(x)-f(y)| \leq |x-y| sup |f^{'}(t)| \) holds for all \(x, y \in I, x < y\)

Answer 1

By the mean value theorem there is c in I such tthat
\[f(x) - f(y)= f'(c)(x-y)\\
|f(x) - f(y)|=| f'(c)||(x-y)| \le M |x-y|\\


\]
Where M is the bound of f' on I.

Answer 2

thank you very much Mr Elias

Answer 3

yw