Question

When you're determining if a function is continuous or not using limits, how do you know if it's a jump discontinuity or removable discontinuity without graphing?

Answer 1

A removable discontinuity is one you can "fix" by simply adding a point to the function. So it looks like a hole when you graph the function. On the other hand a jump discontinuity cannot be fixed this way because adding a point will not change the fact that the left and right limits are different.

Answer 2

A simple example of a jump discontinuity is just
$$f(x)=\left\{\begin{array}{rcl}-1 & : & x \leq 0 \\ 1 & : & x > 0\end{array}\right.$$