Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. Prove that $\lim_{x \rightarrow 0} \frac{1}{x}\int_0^x f(t) dt = f(0)$.
I'm having a little confusion about proving this. So far, it is clear that $f$ is continuous at 0 and $f$ is Riemann integrable. So with that knowledge, I am trying to use the definition of continuity. So $|\frac{1}{x}\int_0^x f(t) dt - f(0)|=|\frac{1}{x}(f(x)-f(0))-f(0)|$. From here, I'm not sure where to go. Any help is appreciated. Thanks in advance.