Removable Discontinuities (College Board AP® Calculus AB): Revision Note

Removable discontinuity

What is a removable discontinuity?

• A removable discontinuity is a discontinuity in a function that can be 'removed'

  • This is done in order to make the function continuous over an interval containing the discontinuity

• Recall that a function $f$ is continuous at the point $x=c$ if

  • $f\left(c\right)$ exists,

  • $\underset{x\to c}{\lim }f\left(x\right)$ exists,

  • and $\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right)$

• A removable discontinuity is a hole in the function

  • It is a point where:

    • $\underset{x\to c}{\lim }f\left(x\right)$ exists

    • but $f\left(c\right)$ doesn't exist or is not equal to $\underset{x\to c}{\lim }f\left(x\right)$

• For example, $g\left(x\right)=\frac{x}{x}$ is not continuous at $x=0$ because $g\left(0\right)$ doesn't exist

  • However the discontinuity at $x=0$ is removable because $\underset{x\to 0}{\lim }g\left(x\right)=1$

    • I.e. the limit does exist at the point of discontinuity

Removing discontinuities

How can I remove a removable discontinuity?

• To remove a removable discontinuity the function needs to be redefined

  • This is done by defining $f\left(c\right)$ to be equal to $\underset{x\to c}{\lim }f\left(x\right)$ at the point of discontinuity

  • The function then 'ticks all the boxes' to be continuous at that point

• For example, if we instead define the function $g$ by $g\left(x\right)=\{\begin{array}{l}\frac{x}{x},x\ne 0\\ 1,x=0\end{array}$

  • Then $g$ becomes continuous over all the real numbers, including $x=0$

  • We have removed the removable discontinuity!