Is a function $f \in \mathbb{C}^{ \infty}[0,l]$ always in $L^2(0,l)$?

Question

I was trying to find a function that is not in $L^2(0,l)$ but that it is in $\mathbb{C}^{\infty}[0,l]$ for l>0.

But if the function is continuous at both sides of the interval then it is integrable, isn't it?