12.3 Continuity

Arizona is known for its dry heat. On a particular day, the temperature might rise as high as ${118}^{\circ }\text{F}$ and drop down only to a brisk ${95}^{\circ }\text{F}\text{.}$ [link] shows the function $T,$ where the output of $T\left(x\right)$ 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.

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 ${96}^{\circ }\text{F}$ . By 2 p.m. the temperature had risen to ${116}^{\circ }\text{F,}$ and by 4 p.m. it was ${118}^{\circ }\text{F}\text{.}$ Sometime between 2 a.m. and 4 p.m. , the temperature outside must have been exactly ${110.5}^{\circ }\text{F}\text{.}$ In fact, any temperature between ${96}^{\circ }\text{F}$ and ${118}^{\circ }\text{F}$ occurred at some point that day. This means all real numbers in the output between ${96}^{\circ }\text{F}$ and ${118}^{\circ }\text{F}$ 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 $D\left(x\right)$ 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.

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.