The easiest way to see points of discontinuity is to graph the function in which you are interested. The trig functions are all continuous except at zeroes of the denominator, e.g. since tan x = sin x/cos x, tan will be discontinuous where cos x =0.
sec x = 1/cos x ... discontinuities will be in the same places where tan x is discontinuous, namely at zeroes of cos x.
The function you wrote as tan^-1(5x) is unclear. Did you mean the arctan(5x) or 1/tan(5x)?