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

Jump discontinuity

What is a jump discontinuity?

• A jump discontinuity 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 $f\left(x\right)=\{\begin{array}{l}{x}^3-x^2-2x,x<1\\ 8-x^2-3x,x\ge 1\end{array}$

  • $\underset{x\to 1^{-}}{\lim }f\left(x\right)=-2$

  • $\underset{x\to 1^{+}}{\lim }f\left(x\right)=4$

  • So $f$ has a jump discontinuity at $x=1$

Essential discontinuity

• 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 $f\left(x\right)=\{\begin{array}{l}x-3,x\le 5\\ 4-\frac{1}{x-5},x>5\end{array}$

  • $\underset{x\to 5^{-}}{\lim }f\left(x\right)=2$ (and $f\left(5\right)=2$ as well)

  • But $\underset{x\to 5^{+}}{\lim }f\left(x\right)=-\infty$

  • Because the right-hand limit is infinite, $f$ has an essential discontinuity at $x=5$