Suppose $g\in L^1([0,\infty))$ and $\int_0^\infty g(x)\, dx=1$.
How to prove that if $f:[0,\infty) \rightarrow \mathbb{R}$ is a continuous function then $n \int_0^1 f(x+t) g(nt)\, dt \rightarrow f(x)$ as $n\rightarrow \infty$ for $x\in \mathbb{R}$?
Say $nt=u$ in the integral then we get $\int_0^n f(x+u/n) g(u) \,du$. Estimating this integral from above and letting $f_n(x)= f(x+u/n) \chi_{[0,n]} $ gives nothing since we don't know whether $f$, being pointwise limit of this sequence, is integrable on $\mathbb{R}$. How can I use the continuity of $f$? Somehow Lebesgue Dominated Convergence Theorem is gonna play a role but I cannot get it, could you help?
Obrigado.