I consider the following periodic function $f : \mathbb{R}^d \rightarrow \mathbb{R}$, given by $$f(x) = \sum_{n \in \mathbb{Z}^d \backslash\{0\}} \frac{\mathrm{e}^{\mathrm{i} \langle x, n\rangle} }{\lVert n \rVert^d},$$ where $\langle \cdot , \cdot \rangle$ is the usual $d$-dimensional scalar product.
Question: Is it true that $f$ is not a continuous function?
Remark: In dimension $d=1$, the function $f$ has a closed form expression, see this question: Function with Fourier coefficient $1 / \lvert n \rvert$, $n \neq 0$? We see that the function is discontinuous (and actually diverges) around the origin. My guess is that something similar is happening in any dimension $d\geq 1$.