Is $\frac{1}{\phi(b)} \sum_{1 \leq x \leq b, (x, b) = 1} e^{-\pi i a x / b}$ continuous in $a$ and $b$?

Question

Let \[ f(a, b) = \frac{1}{\phi(b)} \sum_{1 \leq x \leq b, (x, b) = 1} e^{-\pi i a x / b}, \]

with $a < b$, $(a, b) = 1$.

Then is $f(a, b)$ continuous in $a/b \in \mathbb{Q}$ ?

This question just aroze during an excursion into some problem.

Thanks in advance.