Types Of Discontinuities

Types of Discontinuities

When a function fails to be continuous at a point $x = a$, it falls into one of three categories based on how the three continuity conditions break down.

Removable Discontinuities

A removable discontinuity occurs when $\lim_{x\to a} f(x)$ exists, but either $f(a)$ is undefined or $f(a) \neq \lim_{x\to a} f(x)$. Graphically, this appears as a hole in the graph. The function behaves smoothly everywhere nearby—the discontinuity could be "fixed" by simply defining or redefining $f(a)$ to equal the limit.

Example: For $f(x) = \dfrac{x^2 - 4}{x - 2}$ at $x = 2$, the function is undefined at $x = 2$, but we can simplify: $f(x) = x + 2$ for $x \neq 2$. Thus $\lim_{x\to 2} f(x) = 4$. Setting $f(2) = 4$ would remove the discontinuity.

Jump Discontinuities

A jump discontinuity happens when both one-sided limits exist and are finite—$\lim_{x\to a^-} f(x)$ and $\lim_{x\to a^+} f(x)$ both exist—but they are not equal. The function abruptly 'jumps' from one value to another as $x$ crosses $a$. This is common in step functions and piecewise-defined functions.

Infinite Discontinuities

An infinite discontinuity occurs when at least one one-sided limit is $+\infty$ or $-\infty$. This creates a vertical asymptote at $x = a$. The function shoots off to infinity rather than settling to a finite height.

Classification Quick Reference

To identify the type: first check whether the two-sided limit exists. If yes, compare it to $f(a)$ (removable if they mismatch). If the limit doesn't exist, determine whether one-sided limits are unequal finite numbers (jump) or one is infinite (infinite discontinuity).