Theorem: Let $f_\epsilon = f* K_{\epsilon}$, where $K\in L^1(\mathbb R^n)$ and $\int_{\mathbb R^n} K =1$. If $f\in L^p(\mathbb R^n)$, then $\|f_{\epsilon} -f \|_p\rightarrow 0$ as $\epsilon \rightarrow 0$.
($K_\epsilon(x) := \epsilon^{-n}K(x/\epsilon)$, $f*K(x) = \int f(t)K(x-t)\, dt$ is its convolution.)
Prove that the above theorem is wrong when $p = \infty$.
I'm not sure what easy example to think, help please.
Suppose that $n = 1$, and suppose that $K_\epsilon (x) = \epsilon^{-1} \mathbf 1_{[-\epsilon / 2, \epsilon / 2]}$. Then for any function $f \in L^\infty(\mathbb R)$ and any point $x \in \mathbb R$, $f_\epsilon(x)$ is equal to the average of $f$ over an interval of width $\epsilon$ centred at $x$.
Now consider the function $f(x) = \mathbf 1_{[0, \infty)}$. Can you write down an expression for the convolution $f_\epsilon(x)$? Can you show that the convolution takes the value $\frac 1 2$ at $x = 0$, and hence, $\| f_\epsilon - f\|_\infty = \frac 1 2$ for all $\epsilon > 0$?