I want to prove that if $f$ is Lipschitz and is assumed to be continuous on some interval $[a,b]$ and differentiable on $(a,b)$, then $f'$ is bounded on $(a,b)$.
I know that because $|f(x)-f(y)|$ $\leq$ $M|x-y|$,
I can say that $\left \vert \frac{f(x)-f(y)}{x-y}\right \vert\leq M$.
The next step is where I get lost. Why does it then follow that for all $x \in [a,b]$, $\left \vert\lim_{(x→y)} \frac{f(x)−f(y)}{(x-y)}\right\vert≤M$? Is that just a basic assumption that I can make?
Thanks!
On
Lipschitz functions are always continuous (by definition, you have their modulus of continuity).
Derivatives exists only Lebesgue almost everywhere. For details, you should take a look to Rademacher theorem (see for istance Evans Gariepy, Measure theory and fine properties of functions). To get bounds of incremental ratios you could use $\limsup$ and $\liminf$ with trivial bounds given by your Lipschitz condition.
By the Lipschitz condition, $$ \frac{|f(x) -f(y) |}{|x-y|} \leq M$$ for all $x,y \in [a,b]$. This implies that as long as the limit in question exists, it will be $\leq M$.