# Introduction to Calculus

## Continuity

Arizona is known for its dry heat. On a particular day, the temperature might rise as high as 
  and drop down only to a brisk 
   shows the function 
  where the output of 
  is the temperature in Fahrenheit degrees and the input 
  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  indicates that, at 2 a.m., the temperature was 
 . By 2 p.m. the temperature had risen to 
  and by 4 p.m. it was 
  Sometime between 2 a.m. and 4 p.m., the temperature outside must have been exactly 
  In fact, any temperature between 
  and 
  occurred at some point that day. This means all real numbers in the output between 
  and 
  are generated at some point by the function according to the intermediate value theorem,
Look again at . 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 
  where 
  is the output representing cost in dollars for parking 
  number of hours. See .
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.
So how can we decide if a function is continuous at a particular number? We can check three different conditions. Let’s use the function 
  represented in  as an example.
Condition 1 According to Condition 1, the function 
  defined at 
  must exist. In other words, there is a y-coordinate at 
  as in .
Condition 2 According to Condition 2, at 
  the limit, written 
  must exist. This means that at 
  the left-hand limit must equal the right-hand limit. Notice as the graph of 
  in  approaches 
  from the left and right, the same y-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at 
 .
Condition 3 According to Condition 3, the corresponding 
  coordinate at 
  fills in the hole in the graph of 
  This is written 
  
Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented in  so the function is continuous as 
  
 through  provide several examples of graphs of functions that are not continuous at 
  and the condition or conditions that fail.

### Identifying a Jump Discontinuity

Discontinuity can occur in different ways. We saw in the previous section that a function could have a left-hand limit and a right-hand limit even if they are not equal. If the left- and right-hand limits exist but are different, the graph “jumps” at 
 . The function is said to have a jump discontinuity.
As an example, look at the graph of the function 
  in . Notice as 
  approaches 
  how the output approaches different values from the left and from the right.

### Identifying Removable Discontinuity

Some functions have a discontinuity, but it is possible to redefine the function at that point to make it continuous. This type of function is said to have a removable discontinuity. Let’s look at the function 
  represented by the graph in . The function has a limit. However, there is a hole at 
 . The hole can be filled by extending the domain to include the input 
  and defining the corresponding output of the function at that value as the limit of the function at 
 .

### Recognizing Continuous and Discontinuous Real-Number Functions

Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of 
  as 
  approaches 
  is the same as the value of 
  when 
  So 
  There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.

### Determining the Input Values for Which a Function Is Discontinuous

Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A piecewise function may have discontinuities at the boundary points of the function as well as within the functions that make it up.
To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that polynomial functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function.

### 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.

### Key Concepts

1. A continuous function can be represented by a graph without holes or breaks.
2. A function whose graph has holes is a discontinuous function.
3. A function is continuous at a particular number if three conditions are met:
    


4. A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
5. A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See .
6. Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See  and .
7. 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  and .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, determine why the function 
  is discontinuous at a given point 
  on the graph. State which condition fails.
For the following exercises, determine whether or not the given function 
  is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

### Graphical

For the following exercises, refer to . Each square represents one square unit. For each value of 
  determine which of the three conditions of continuity are satisfied at 
  and which are not.
For the following exercises, use a graphing utility to graph the function 
  as in . Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.
For the following exercises, consider the function shown in .