CK-12-Pre-Calculus Concepts

1.10. Continuity and Discontinuity http://www.ck12.org

f(x) =







x^2 − 4 x< 1

− 1 x= 1

−^12 x+ 1 x> 1

Solution: There is a jump discontinuity atx=1. The piecewise function describes a function in three parts; a
parabola on the left, a single point in the middle and a line on the right.
Example C
Identify the discontinuity of the function below.

Solution:Since there is a vertical asymptote atx=1, this is an infinite discontinuity.
Concept Problem Revisited
There are three ways that functions can be discontinuous. When a rational function has a vertical asymptote as a
result of the denominator being equal to zero at some point, it will have an infinite discontinuity at that point. When
the numerator and denominator of a rational function have one or more of the same factors, there will be removable
discontinuities corresponding to each of these factors. Finally, when the different parts of a piecewise function don’t
“match”, there will be a jump discontinuity.

Vocabulary

Removable discontinuitiesare also known as holes. They occur when factors can be algebraically canceled from
rational functions.
Jump discontinuitiesoccur most often with piecewise functions when the pieces don’t match up.
Infinite discontinuitiesoccur when a factor in the denominator of the function is zero.

Guided Practice

1. Describe the continuity or discontinuity of the functionf(x) =sin(^1 x).