I want to write the following equality: $$\lim_{h\to0}\left[\int\limits _{a}^{b}\frac{1}{h}\left[g\left(x-y+h\right)-g\left(x-y\right)\right]f\left(y\right)dy\right]=\left[\int\limits _{a}^{b}\lim_{h\to0}\frac{1}{h}\left[g\left(x-y+h\right)-g\left(x-y\right)\right]f\left(y\right)dy\right]$$
But i'm not sure what are sufficient terms for it to be correct and how to properly justify it. In my specific case $f$ is continuous and $g$ is continuously differentiable.