Types of Discontinuity

When a function fails the three-condition test of §1 at a point, the way it fails classifies the discontinuity. There are three standard types met at this level.

Removable discontinuity. Here the two-sided limit $\displaystyle\lim_{x \to a} f(x)$ exists (both one-sided limits are finite and equal), but it either does not equal $f(a)$, or $f(a)$ is undefined. The graph has a single "hole" that could be repaired by simply redefining (or defining) $f(a)$ to equal the limit — hence removable. This is the type that arises whenever a rational expression has a common factor that cancels, such as $\dfrac{x^2-4}{x-2} = x+2$ for $x \neq 2$: the limit at $2$ is $4$, but the original function is undefined (or wrongly defined) there.

Jump (finite) discontinuity. Here the left-hand limit and the right-hand limit both exist and are finite, but are not equal to each other, so the two-sided limit does not exist. The graph "jumps" from one level to another across $x = a$; the size of the jump is $\left|\displaystyle\lim_{x \to a^+} f(x) - \lim_{x \to a^-} f(x)\right|$. No single redefinition of $f(a)$ can repair a jump, because the two sides simply do not meet. Jump discontinuities are typical of piecewise functions whose pieces have mismatched values at the joining point.

Infinite discontinuity. Here at least one of the one-sided limits is infinite (the function grows without bound), so no finite limit exists. This is the behaviour of a rational function at a point where the denominator is zero but the numerator is not, for example $\dfrac{1}{x-3}$ as $x \to 3$, where the graph shoots off along a vertical asymptote.

Summary table.