1.10. Continuity and Discontinuity http://www.ck12.org
1.10 Continuity and Discontinuity
Here you will learn the formal definition of continuity, the three types of discontinuities and more about piecewise
functions.
Continuity is a property of functions that can be drawn without lifting your pencil. Some functions, like the reciprocal
functions, have two distinct parts that are unconnected. Functions that are unconnected are discontinuous. What are
the three ways functions can be discontinuous and how do they come about?
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Guidance
Functions that can be drawn without lifting up your pencil are called continuous functions. You will define continu-
ous in a more mathematically rigorous way after you study limits.
There are three types of discontinuities: Removable, Jump and Infinite.
Removable discontinuitiesoccur when a rational function has a factor with anxthat exists in both the numerator
and the denominator. Removable discontinuities are shown in a graph by a hollow circle that is also known as a hole.
Below is the graph forf(x) =(x+^2 x)(+ 1 x+^1 ). Notice that it looks just likey=x+2 except for the hole atx=−1. There
is a hole atx=−1 because whenx=− 1 ,f(x) =^00.
Removable discontinuities can be “filled in” if you make the function a piecewise function and define a part of the
function at the point where the hole is. In the example above, to makef(x)continuous you could redefine it as:
f(x) ={(x+ 2 )(x+ 1 )
x+ 1 , x^6 =−^1
1 , x=− 1