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In this section, you will:
  • Determine whether a function is continuous at a number.
  • Determine the numbers for which a function is discontinuous.
  • Determine whether a function is continuous.

Arizona is known for its dry heat. On a particular day, the temperature might rise as high as 118 F and drop down only to a brisk 95 F . [link] shows the function T , where the output of T ( x ) is the temperature in Fahrenheit degrees and the input x is the time of day, using a 24-hour clock on a particular summer day.

Graph of function that maps the time since midnight to the temperature. The x-axis, labelled x, represents the hours since midnight from 0 to 24. The y-axis, labelled T(x), represents the temperature from 0 to 120. The function is continuous that peaks at (16, 118).
Temperature as a function of time forms a continuous function.

When we analyze this graph, we notice a specific characteristic. There are no breaks in the graph. We could trace the graph without picking up our pencil. This single observation tells us a great deal about the function. In this section, we will investigate functions with and without breaks.

Determining whether a function is continuous at a number

Let’s consider a specific example of temperature in terms of date and location, such as June 27, 2013, in Phoenix, AZ. The graph in [link] indicates that, at 2 a.m. , the temperature was 96 F . By 2 p.m. the temperature had risen to 116 F, and by 4 p.m. it was 118 F . Sometime between 2 a.m. and 4 p.m. , the temperature outside must have been exactly 110.5 F . In fact, any temperature between 96 F and 118 F occurred at some point that day. This means all real numbers in the output between 96 F and 118 F are generated at some point by the function according to the intermediate value theorem,

Look again at [link] . There are no breaks in the function’s graph for this 24-hour period. At no point did the temperature cease to exist, nor was there a point at which the temperature jumped instantaneously by several degrees. A function that has no holes or breaks in its graph is known as a continuous function    . Temperature as a function of time is an example of a continuous function.

If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let’s create the function D , where D ( x ) is the output representing cost in dollars for parking x number of hours. See [link] .

Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a $25 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is $16. We can never be charged $13, $14, or $15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function’s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.

Graph of function that maps the time since midnight to the temperature. The x-axis represents the hours parked from 0 to 24. The y-axis represents dollars amounting from 0 to 28. The function is a step-function.
Parking-garage charges form a discontinuous function.

A function that remains level for an interval and then jumps instantaneously to a higher value is called a stepwise function . This function is an example.

A function that has any hole or break in its graph is known as a discontinuous function    . A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function.

Questions & Answers

Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
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
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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