A function is continuous at a point if its limit exists at that point.
I think it's false because of the counterexample of one-sided limits?
when we say "a limit exists" it usually implies the limit from both sides. If not, mention is made as to the side.
However, even as it stands it is false because you might have a rational function
f(x) = (x^2-4)/(x-2)
At every point except x=2, f is continuous. The limit of f(x) at x=2 is 4, but f is not defined at x=2, so it is not continuous there.