Suppose $g\in L^1([0,\infty))$ and $f$ is measurable. If there exists $M:=\lim\_{n\to \infty } \int \_0 ^n f(x)dx$, then 
$$\lim\_{c\to0}\int \_0 ^\infty f(x)g(cx)dx = Mg(0) $$

In order to use dominated convergence theorem, I tried to find $F \in L^1([0,\infty))$ s.t., 
$$\lvert f(x)(g(cx)-g(0)\rvert=\left\lvert f(x)\int \_0 ^{cx} g'(t)dt\right\rvert\leq F.$$
But I couldn't find it.