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Arizona is known for its dry heat. On a particular day, the temperature might rise as high as $\text{\hspace{0.17em}}{118}^{\circ}\text{F}\text{\hspace{0.17em}}$ and drop down only to a brisk $\text{\hspace{0.17em}}{95}^{\circ}\text{F}\text{.}\text{\hspace{0.17em}}$ [link] shows the function $\text{\hspace{0.17em}}T,$ where the output of $\text{\hspace{0.17em}}T\left(x\right)\text{\hspace{0.17em}}$ is the temperature in Fahrenheit degrees and the input $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the time of day, using a 24-hour clock on a particular summer day.
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.
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 $\text{\hspace{0.17em}}{96}^{\circ}\text{F}$ . By 2 p.m. the temperature had risen to $\text{\hspace{0.17em}}{116}^{\circ}\text{F,}\text{\hspace{0.17em}}$ and by 4 p.m. it was $\text{\hspace{0.17em}}{118}^{\circ}\text{F}\text{.}\text{\hspace{0.17em}}$ Sometime between 2 a.m. and 4 p.m. , the temperature outside must have been exactly $\text{\hspace{0.17em}}{110.5}^{\circ}\text{F}\text{.}\text{\hspace{0.17em}}$ In fact, any temperature between $\text{\hspace{0.17em}}{96}^{\circ}\text{F}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{118}^{\circ}\text{F}\text{\hspace{0.17em}}$ occurred at some point that day. This means all real numbers in the output between $\text{\hspace{0.17em}}{96}^{\circ}\text{F}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{118}^{\circ}\text{F}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}D,$ where $\text{\hspace{0.17em}}D\left(x\right)\text{\hspace{0.17em}}$ is the output representing cost in dollars for parking $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ 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.
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.
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