Let $f: [a,b] \to \mathbb{R}$ be a differentiable function with continuous derivative. Prove the existence of $>0$ such that for all $x, y\in[a, b]$ the inequality |()−()|≤ |−| is true.
I was thinking of proving this using the Mean Value Theorem but I am not really sure how to proceed. Any help is appreciated.
The derivative is continuous on the compact set $[a,b]$ and hence bounded by some M. Now use MVT
hint
$ f $ is continuous and differentiable at $ [a,b] \implies$
$$(\forall x,y\in[a,b])\;(\exists c\in(x,y))\;:$$ $$\; f(x)-f(y)=(x-y)f'(c)$$
but $ f' $ is continuous at the compact $ [a,b] $ $$\implies f' \text{ is bounded}$$
$$\implies (\exists M>0)\;:$$ $$\;(\forall t\in [a,b])\;|f'(t)|\le M$$
