Let $f$ be integrable, continuous, and vanishing over $\mathbb{R}$, i.e. $f \in L_1(\mathbb{R}) \cap \mathcal{C}_0(\mathbb{R})$. We define $$ g = \sum_{n\in \mathbb{Z}} f(\cdot - n )$$ the periodization of $f$ with period $1$. Is it true that $g \in L_1(\mathbb{T}) \cap \mathcal{C}(\mathbb{T})$?
Note: $g$ can be defined as the limit of the Cauchy sequence $g_N = \sum_{|n| \leq N} f(\cdot-n)$ in $L_1(\mathbb{T})$ and the series of functions indeed converges, at least in $L_1(\mathbb{T})$. My question is therefore to know if the limit is necessarily a continuous periodic function, using the continuity of $f$ and the fact that it vanishes.