how is the signum function neither continuous nor discontinuous at $x=0$. In my book, there is a sentence that says exactly this:

"The function $\mathrm{sgn}(x)= \dfrac{x}{|x|}$ is neither continuous nor discontinuous at $x=0$. How is this possible?"

It was easy for me to tell it is not continuous at $x=0$ as there is no limit existence due to left-right limit inequality, or simply because the graph is broken at $x=0$. But I can't understand the second part, which claims it is also not discontinuous.

I've always thought that "if it is not continuous, then discontinuous", but apparently it seems to be wrong. How is this function not discontinuous?

source: A complete course: calculus(8th edition)
A function cannot be continuous and discontinuous at the same point.

Yes the function is discontinuous which is right as per your argument.

I think the question wanted to convey this..

It has a jumped discontinuity which means if the function is assigned some value at the point of discontinuity it cannot be made continuous.

But the function is definitely discontinuous at x=0.
the sgn function is a discontinuous function (isolated/jump discontinuity). it will be continuous if domain of the function is changed to R- or R+.

You can get some info about the types of discontinuity from the following sources:
https://en.wikipedia.org/wiki/Continuous_functio
https://en.wikipedia.org/wiki/Classification_of_discontinuities#Removable_discontinuity