Prove convergence to zero of $f(t + x) - f(x)$ in the $L^p$-norm

Question

I'd like to show that $\Vert f(t + x) - f(x) \Vert_p \rightarrow 0$ as $t \rightarrow 0$, where $f \in L^p([0, \infty))$ and $\Vert \cdot \Vert_p$ is the usual norm on $L^p$. First I thought of using the Lebesgue-Dominated-Convergence theorem but I can't quite see how to estimate the function $f(t + x) - f(x)$.

Any hint is appreciated!

Thanks!

Answer 1

Hint. Use that $C^0_c([0,\infty))$ is dense in $L^p([0,\infty))$ and prove your claim first for a continuous function with compact support (by dominated convergence) and than use the denseness and approximate an arbritrary $f \in L^p$ by an $g \in C^0_c$.