2.3 Types of Discontinuities
Notes
There are three main types.
- Removable: the limit exists but differs from $f(a)$ or $f(a)$ is undefined (a hole). Example: $\displaystyle \frac{x^2 - 9}{x - 3}$ at $x = 3$.
- Jump: the left and right limits both exist but are different.
- Infinite: the function grows without bound near $a$ (a vertical asymptote). Example: $\displaystyle \frac{1}{(x - 1)^2}$ at $x = 1$.
- A removable discontinuity can be "fixed" by redefining $f(a)$ as the limit.
Practice
What type of discontinuity does $\displaystyle f(x) = \frac{x^2 - 9}{x - 3}$ have at $x = 3$?
It simplifies to $x + 3$ with a hole at $x = 3$, and the limit is 6.
Let $f(x) = x + 1$ for $x < 2$ and $f(x) = x - 1$ for $x \geq 2$. What happens at $x = 2$?
The left limit is 3 and the right limit is 1. Both exist but they differ.
What type of discontinuity does $\displaystyle f(x) = \frac{1}{(x - 1)^2}$ have at $x = 1$?
The function grows without bound near 1, giving a vertical asymptote.
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