Based on a few online searches and the analogue of mollifiers, I think the following should be true. However I was not able to find a proof. Could you please provide a proof or reference? Thanks.
Let $f : \mathbb{R} \to \mathbb{R}$ be a continous bounded function supported on some finite interval of the real numbers. For $\delta > 0$, let $\phi_\delta(x) = \delta^{-1}\phi(x/\delta)$, where $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ is the Gaussian. Let $f_\delta$ be the convolution $f \ast \phi_\delta$. Is it true that $f_\delta \to f$ pointwise as $\delta \to 0^+$?