Does tan(x) have points of discontinuity?

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.