# How to Determine Whether a Function Is Discontinuous

As your pre-calculus teacher will tell you, functions that aren't continuous at an *x* value either have a *removable discontinuity* (a hole in the graph of the function) or a *nonremovable* *discontinuity *(such as a jump or an asymptote in the graph)*:*

**If the function factors and the bottom term cancels, the discontinuity at the***x*-value for which the denominator was zero is removable, so the graph has a hole in it.For example, this function factors as shown:

After canceling, it leaves you with

*x*– 7. Therefore*x*+ 3 = 0 (or*x*= –3) is a removable discontinuity — the graph has a hole, like you see in Figure a.The graph of a removable discontinuity leaves you feeling empty, whereas a graph of a nonremovable discontinuity leaves you feeling jumpy.**If a term doesn't cancel, the discontinuity at this***x*value corresponding to this term for which the denominator is zero is nonremovable, and the graph has a vertical asymptote.The following function factors as shown:

Because the

*x*+ 1 cancels, you have a removable discontinuity at*x*= –1 (you'd see a hole in the graph there, not an asymptote). But the*x*– 6 didn't cancel in the denominator, so you have a nonremovable discontinuity at*x*= 6. This discontinuity creates a vertical asymptote in the graph at*x*= 6. Figure b shows the graph of*g*(*x*).