How to Remove Discontinuities?
A removable discontinuity is a point on a graph that is not defined or does not match the rest of the graph. There is a gap in the chart at that location. Here you will learn more about removing discontinuities

A discontinuity at \(c\) is called removable when the two-sided limit exists at \(c\) but isn’t equal to \(f(c)\). These types of discontinuities can be found by identifying values of the independent variable that are zeros for factors of the denominator that are also factors of the numerator of the given function. The function can be made continuous at these values by redefining \(f(c)\).
Related Topics
A step-by-step guide to removing discontinuities
If there is a discontinuity at \(x=c\) it is said to be removable if
\(lim_{x\to c}f(x)\) exists. Let’s call it \(L\).
However, \(L≠f (c)\) (Either because \(f(c)\) is a different number than \(L\) or because \(f(c)\) has not been defined.)
By introducing a new function, say \(g\), we can “delete” the discontinuity \((x)\):
\(\color{blue}{g(x)=}\)\(\color{blue}{\begin{cases} f(x) \quad if\:\ x ≠ c \\ L \quad if\:\ x=c \end{cases}}\)
We now have \(g(x)=f(x)\) for all \(x≠c\) and \(g\) is continuous at \(c\).
Removing Discontinuities – Example 1:
The function \(f(x)=\frac{x^2-1}{x-1}\) is discontinuous at \(x=1\). Find what value should be assigned to \(f(1)\) to make\(f(x)\) continuous.
\(f(x)=\frac{x^2-1}{x-1}\) is discontinuous at \(x=1\) becuse \(f(1)\) does not exist.
But \(lim_{x\to 1}\frac{x^2-1}{x-1}= lim_{x\to 1}\frac{(x-1)(x+1)}{x-1}=x+1=1+1=2\)
So we remove the discontinuity by defining:
\(g(x)=\)\(\begin{cases} \frac{x^2-1}{x-1} \quad if\:\ x ≠ 1 \\ 2 \quad \ if\:\ x=1 \end{cases}\)
Removing Discontinuities – Example 2:
The function \(f(x)=\frac{sin x}{x}\) is discontinuous at \(x=0\). Find what value should be assigned to \(f(0)\) to make\(f(x)\) continuous.
The function is not defined in \(x=0\). As \(sin x\) is continuous at every \(x\), then the initial function \(f(x)=\frac{sin x}{x}\) is continuous for all \(x\) except the point \(x=0\).
\(g(x)=\)\(\begin{cases} \frac{sin x}{x} \quad if\:\ x ≠ 0 \\ 1 \quad \ if\:\ x=0 \end{cases}\)
Exercises for Removing Discontinuities
In each of the following functions, remove the discontinuity at the given point.
- \(\color{blue}{f(x)=\frac{x^3+64}{x+4}}\), \(\color{blue}{a=-4}\)
- \(\color{blue}{f(x)=\frac{2-\sqrt{x}}{4-x}}\), \(\color{blue}{a=4}\)
- \(\color{blue}{f(x)=\frac{5x^2+4x-1}{x+1}}\), \(\color{blue}{a=-1}\)
- \(\color{blue}{f(x)=\frac{9x^2-4}{3x+2}}\), \(\color{blue}{a=-\frac{2}{3}}\)

- \(\color{blue}{g(x)=}\)\(\color{blue}{\begin{cases} \frac{x^3+64}{x+4} \quad if\:\ x ≠ -4 \\ 48 \quad if\:\ x=-4 \end{cases}}\)
- \(\color{blue}{g(x)=}\)\(\color{blue}{\begin{cases} \frac{2-\sqrt{x}}{4-x} \quad if\:\ x ≠ 4 \\ \frac{1}{4} \quad if\:\ x=4 \end{cases}}\)
- \(\color{blue}{g(x)=}\)\(\color{blue}{\begin{cases} \frac{5x^2+4x-1}{x+1} \quad if\:\ x ≠ -1 \\ -6 \quad if\:\ x=-1 \end{cases}}\)
- \(\color{blue}{g(x)=}\)\(\color{blue}{\begin{cases} \frac{9x^2-4}{3x+2} \quad if\:\ x ≠ -\frac{2}{3} \\ -4 \quad if\:\ x=-\frac{2}{3} \end{cases}}\)
Related to This Article
More math articles
- How to Solve Systems of Linear Inequalities?
- TExES Core Subjects Math Formulas
- Top 10 Tips to Create the HSPT Math Study Plan
- Top 5 Free Websites for OAR Math Preparation
- How to Translate Phrases into an Algebraic Statement? (+FREE Worksheet!)
- Quotient Quest: How to Estimate Division with Two-Digit Divisors
- The Ultimate KAP Algebra 1 Course (+FREE Worksheets)
- The Best Calculators for Grade 9 Students
- Ratio, Proportion and Percentages Puzzle -Critical Thinking 9
- How to Find Discriminant of Quadratic Equation?
What people say about "How to Remove Discontinuities? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.