Convolution with mollifier $f\star \rho_n\not\to f$? for $f\in L^{\infty}$

Question

I just learned that if $\rho_n$ is a sequence of molifiers and $f$ a function in $L^p$ with $p\neq \infty$ than the convolution $f\star \rho_n\to f$. But what about $p=\infty$? I there an easy example $f\in L^{\infty}$ such that $f\star \rho_n\not\to f$?

Answer 1

Yes, take any $f \in L^{\infty}$ which is not continuous.

To see this, observe that each $f \ast \rho_n$ is continuous and $f \ast \rho_n \to f$ in $L^{\infty}$ implies the limit $f$ must also be continuous by the uniform limit theorem.

Note that you will still have convergence in $L^p_{\mathrm{loc}}$ for all $p < \infty$ and convergence almost everywhere (including at all points of continuity), but the convergence can never be uniform if $f$ is discontinuous.