# 12.3 Continuity  (Page 5/10)

 Page 5 / 10

Determine where the function is discontinuous.

$x=6$

## Determining whether a function is continuous

To determine whether a piecewise function    is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.

Given a piecewise function, determine whether it is continuous.

1. Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied?
2. For each boundary point $\text{\hspace{0.17em}}x=a\text{\hspace{0.17em}}$ of the piecewise function, determine if each of the three conditions hold.

## Determining whether a piecewise function is continuous

Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity.

$fx=\left\{\begin{array}{ll}\mathrm{sin}\left(x\right),\hfill & x<0\hfill \\ {x}^{3},\hfill & x>0\hfill \end{array}$

The two functions composing this piecewise function are $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(x\right)\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}x<0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}x>0.\text{\hspace{0.17em}}$ The sine function and all polynomial functions are continuous everywhere. Any discontinuities would be at the boundary point,

At $\text{\hspace{0.17em}}x=0,$ let us check the three conditions of continuity.

Condition 1:

Because all three conditions are not satisfied at $\text{\hspace{0.17em}}x=0,$ the function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is discontinuous at $\text{\hspace{0.17em}}x=0.$

Access these online resources for additional instruction and practice with continuity.

## Key concepts

• A continuous function can be represented by a graph without holes or breaks.
• A function whose graph has holes is a discontinuous function.
• A function is continuous at a particular number if three conditions are met:
• Condition 1: $\text{\hspace{0.17em}}f\left(a\right)\text{\hspace{0.17em}}$ exists.
• Condition 2: $\text{\hspace{0.17em}}\underset{x\to a}{\mathrm{lim}}f\left(x\right)\text{\hspace{0.17em}}$ exists at $\text{\hspace{0.17em}}x=a.$
• Condition 3: $\text{\hspace{0.17em}}\underset{x\to a}{\mathrm{lim}}f\left(x\right)=f\left(a\right).$
• A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
• A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See [link] .
• Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See [link] and [link] .
• For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See [link] and [link] .

## Verbal

State in your own words what it means for a function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ to be continuous at $\text{\hspace{0.17em}}x=c.$

Informally, if a function is continuous at $\text{\hspace{0.17em}}x=c,$ then there is no break in the graph of the function at $\text{\hspace{0.17em}}f\left(c\right),$ and $\text{\hspace{0.17em}}f\left(c\right)\text{\hspace{0.17em}}$ is defined.

State in your own words what it means for a function to be continuous on the interval $\text{\hspace{0.17em}}\left(a,b\right).$

## Algebraic

For the following exercises, determine why the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is discontinuous at a given point $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ on the graph. State which condition fails.

discontinuous at $\text{\hspace{0.17em}}a=-3$ ; $\text{\hspace{0.17em}}f\left(-3\right)\text{\hspace{0.17em}}$ does not exist

$f\left(x\right)=\frac{{x}^{2}-16}{x+4},a=-4$

removable discontinuity at $\text{\hspace{0.17em}}a=-4$ ; $\text{\hspace{0.17em}}f\left(-4\right)\text{\hspace{0.17em}}$ is not defined

$f\left(x\right)=\frac{{x}^{2}-16x}{x},a=0$

Discontinuous at $\text{\hspace{0.17em}}a=3$ ; $\text{\hspace{0.17em}}\underset{x\to 3}{\mathrm{lim}}f\left(x\right)=3,$ but $\text{\hspace{0.17em}}f\left(3\right)=6,$ which is not equal to the limit.

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
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ismail
Rectangle coordinate
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it depends on the equation
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yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
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give me an example of a problem so that I can practice answering
x³+y³+z³=42
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dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
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explain this