For a Lebesgue integrable function $f$, the map $t\mapsto\int \chi_{A+t}f(x) dx $ is continuous

Question

Suppose that $ f\in L^1(\mathbb{R})$ and $A $ is Borel subset of $R$. Show that the mapping $t$ to $\int \chi_{A+t}f(x) dx $ is continuous from $\mathbb{R}$ to $\mathbb{R}$.

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I try first by taking A as an interval. I think it works in case this case but I could not figure out when A is general Borel set. Please help me out.