Non-removable Discontinuities (College Board AP® Calculus AB)

Study Guide

Written by: Roger B

Reviewed by: Dan Finlay

A jump continuity occurs at a point where the value of a function makes a sudden 'leap' between two values

Where there is a jump discontinuity
- The limit from the left and the limit from the right both exist
- But they are not equal
A jump discontinuity is not a removable discontinuity
For example, consider the function $f$ defined by $f (x) = \{\begin{cases} x^{3} - x^{2} - 2 x, x < 1 \\ 8 - x^{2} - 3 x, x \geq 1 \end{cases}$
- $\lim_{x \to 1^{-}} f (x) = - 2$
- $\lim_{x \to 1^{+}} f (x) = 4$
- So $f$ has a jump discontinuity at $x = 1$

A function has an essential (or infinite) discontinuity at a point where
- the limit from the left or the limit from the right (or both)
  - do not exist
  - or are infinite
A common example is where the graph of a function has a vertical asymptote

An essential discontinuity is not a removable discontinuity
For example, consider the function $f$ defined by $f (x) = \{\begin{cases} x - 3, x \leq 5 \\ 4 - \frac{1}{x - 5}, x > 5 \end{cases}$
- $\lim_{x \to 5^{-}} f (x) = 2$ (and $f (5) = 2$ as well)
- But $\lim_{x \to 5^{+}} f (x) = - \infty$
- Because the right-hand limit is infinite, $f$ has an essential discontinuity at $x = 5$

Last updated: 6 August 2024

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