Continuity and derivative square integrable implies Holder continuity?

Question

Let $f\in\mathcal{C}^{0}$ such that $f'\in\mathcal{L}^{2}[a,b]$. How can i show that $f$ is Hölder?

My attempt: Assuming that i can apply the fundamental theorem of calculus,i was able to find that $f$ is Holder,but i can not prove that in fact,for any $x,y\in[a,b]$ i can apply the theorem on $f'$.

Answer 1

Hints:

1. The derivative $f'$ is also $L^1$ as $[a,b]$ has finite measure.
2. If $f$ is actually everywhere differentiable, then $f$ is absolutely continuous.
3. By fundamental theorem of calculus we have for $a\le x \le y \le b$ $$ |f(y) - f(x)| \le \int_x^y 1 \cdot |f'(t)| \; dt. $$
4. Apply Cauchy Schwarz on the right hand side.