If $\lvert f \rvert$ is continuous at a point, then is $f$ continuous at the point?

Question

The question is written in the title. Let $f$ be a complex-valued function defined on the compact interval $[0, 2\pi]$. If $\lvert f \rvert$ is continuous at a point $x$, then is $f$ itself also continuous at the point? I know the reverse is trivially true but am confused about whether the continuity of $\lvert f \rvert$ is truly equivalent to the continuity of $f$.

Answer 1

Take the function to be $f(x)=1$ when $x \in [0,\pi]$ and $f(x)=-1$ when $x \in (\pi,2\pi]$. The module is clearly continuous at the point $x=\pi$, but what about the function?