Do we have $\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$?

Question

Let $f:\mathbb{R^+}\to\mathbb{R}$ be an integrable function ($f\in L^1(\mathbb{R}^+,\mathbb{R})$). Do we have $$\lim_{h\to 0^+}\int_0^\infty|f(t+h)-f(t)|dt=0$$ ? How can we prove it ?

Answer 1

Hint. It is known that $\mathcal{C}^{\infty}(\mathbb{R})$ maps with compact support are dense in $L^1(\mathbb{R})$. Hence, it suffices to prove the result for such functions. I should add that this is true, since translations are isometries of $L^1(\mathbb{R})$, namely: $$\|f\|_{L^1(\mathbb{R})}=\|f(\cdot+h)\|_{L^1(\mathbb{R})}.$$ Besides, the result holds for such functions, since they are uniformly continuous.