existence of the convolution with a continuously differentiable function

Question

It is well known (using Fubini's theorem) that the convolution of two $f,g\in L^1(\mathbb{R})$ functions is again in $L^1$, and thus $$ f \ast g(x)=\int_\mathbb{R}{f(x-t)g(t)dt}$$ converges for almost all $x$. I wanted to ask: in case $g\in C^1(\mathbb{R})$ is continuously differentiable with a bounded derivative (or even integrable if necessary), does this necessarily imply that the integral defining $f \ast g(x)$ converges for all $x \in \mathbb{R}$? I am trying to find a minimal condition (not strong properties like compact support) for the integral to converge everywhere.