Prove $\lim\limits_{h\to0}\int_a^b\left|f(x+h)-f(x)\right|\,\mathrm dx=0$ Let $f(x)$ be a Riemann integrable function on any finite interval $I\subset\mathbb{R}$. Supposing $f(x)=0$ for $x\notin[a,b]$, show that $$\lim\limits_{h\to0}\int_a^b\left|f(x+h)-f(x)\right|\,\mathrm{d}x=0.$$
How to prove that if not familiar with the fact that Riemann integrable function is almost everywhere continuous?

Hint.
You can prove the result for continuous functions as a continuous function on a compact interval is uniformly continuous.
Then, if $f$ is Riemann-integrable, you can for all $\epsilon >0$ find a step function $h_\epsilon$ such that:
$$ \int_a^b \left\vert f-h_\epsilon  \right\vert < \epsilon$$ 
and finally a function $g_\epsilon$ continuous such that
$$ \int_a^b \left\vert h_\epsilon-g_\epsilon  \right\vert < \epsilon$$
