# Prove that $F$ is continuous function of $t$

Let $p$ and $q$ be conjugate indices and let $f\in L^p(\mathbb R)$ and $g\in L^q(\mathbb R)$. Prove that $F(t)=\int f(x+t)g(x)dx$ is a continuous function of $t$.

My attempt: For any $h \in \mathbb R$, $$|F(t+h)-F(t)|\le \int|f(x+t+h)-f(x+t)||g(x)|dx$$

$$\le \left(\int|f(x+t+h)-f(x+t)|^pdx \right)^{1/p}\left(\int|g(x)|^q dx\right)^{1/q}$$ by Holder's inequality. However, I have no idea how to prove that $\int|f(x+t+h)-f(x+t)|^pdx=\int|f(x+h)-f(x)|^pdx$ goes to $0$ as $h\to 0$. Does anyone have ideas? I appreciate your help!

Let $\tau_h\colon\mathbb L^p\to\mathbb L^p$ be defined by $\tau_h\left(f\right)(x)=f\left(x+h\right)$. We have to show that $\left\lVert \tau_h(f)-f\right\rVert_p$ goes as $h$ goes to zero, which is done here.