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QUESTION
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.
