Example of function such that her Cesaro mean won't converge in $L^2$ norm

Question

I'm looking for an example of function $f: \mathbb{R} \rightarrow \mathbb{R}$ , $f \in L^{1}(-\pi;\pi)$, and $f$ being $2\pi$ periodic, such that the Cesaro Mean $\phi_{N}f$ is such that $\phi_{N}f(a)$ will not converge to $f(a)$ if $f \in L^{2}(-\pi;\pi)$, for $a \in (-\pi;\pi)$. Thank you!