We know when $1\leq p<\infty$ , the mollification function $f^{\epsilon}=\phi_{\epsilon}*f$ for $L^{p}(R^n)$ functions converge to $f$ in $L^{p}$ norm, when $p=\infty$ it might be wrong. But who can disprove it or give a counterexample will help.
Thank you very much!
Note that $f^\varepsilon$ will always be continuous, so a jump discontinuity anywhere will to the trick (the continuous approximation will have to deviate from $f$ by at least $\frac\delta2$ where $\delta$ is the magnitude of the jump) To become concrete, take $f = \chi_{[0,1]}$ the characteristic function of the unit interval.