Question

How do you find the discontinuity of a function?

Answer 1

The answer to that question depends very much on the function.
In general you need to find points `a` in the domain where
the function is defined in some open interval containing `a`, but

it is NOT true that `lim_(xrarra) f(x) = f(a)`. Perhaps because one of the other (or both) fail to exist or because, although they both exist, they are distinct.

The detail vary widely depending on the function.