Question

Does tan(x) have points of discontinuity?

Answer 1

Yes. It has a dicontinuity at every `x` for which `tanx` is not defined.

These are the `x` for which `cos x =0`

That is: `tan x` is discontinuous at every odd multiple of `pi /2`

These point, of course, are not in the domain of `tan x`.

The discontinuities are non-removable, infinite discontiuities.