Continuity: Learn It 1

  • Determine whether a function is continuous at a number.
  • Determine the input values for which a function is discontinuous.
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 [latex]D[/latex], where [latex]D\left(x\right)[/latex] is the output representing cost in dollars for parking [latex]x[/latex] number of hours.

Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a $24 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.

stepwise 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.

discontinuous function

A function that has any hole or break in its graph is known as 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 [latex]y=f\left(x\right)[/latex] shown below as an example.

Graph of an increasing function with a discontinuity at (a, f(a)).

Condition 1 According to Condition 1, the function [latex]f\left(a\right)[/latex] defined at [latex]x=a[/latex] must exist. In other words, there is a y-coordinate at [latex]x=a[/latex].

Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.

Condition 2 According to Condition 2, at [latex]x=a[/latex] the limit, written [latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)[/latex], must exist. This means that at [latex]x=a[/latex] the left-hand limit must equal the right-hand limit. Notice as the graph of [latex]f[/latex] approaches [latex]x=a[/latex] 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 [latex]x=a[/latex] .

Condition 3 According to Condition 3, the corresponding [latex]y[/latex] coordinate at [latex]x=a[/latex] fills in the hole in the graph of [latex]f[/latex]. This is written [latex]\underset{x\to a}{\mathrm{lim}}f\left(x\right)=f\left(a\right)[/latex].

Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented below so the function is continuous as [latex]x=a[/latex].

Graph of an increasing function with filled-in discontinuity at (a, f(a)).
All three conditions are satisfied. The function is continuous at [latex]x=a[/latex] .

Below are several examples of graphs of functions that are not continuous at [latex]x=a[/latex] and the condition or conditions that fail.

Graph of an increasing function with a discontinuity at (a, f(a)).
 Condition 2 is satisfied. Conditions 1 and 3 both fail.

 

Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.
Conditions 1 and 2 are both satisfied. Condition 3 fails.
Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.
Condition 1 is satisfied. Conditions 2 and 3 fail.
Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)) and another increasing segment from (a, f(a) - 1) to positive infinity. This graph does not include the point (a, f(a)).
Conditions 1, 2, and 3 all fail.

definition of continuity

A function [latex]f\left(x\right)[/latex] is continuous at [latex]x=a[/latex] provided all three of the following conditions hold true:

Condition 1: [latex]f(a)[/latex] exists.

Condition 2: [latex]\underset{x\to a}{\mathrm{lim}}f(x)[/latex] exists at [latex]x=a[/latex].

Condition 2: [latex]\underset{x\to a}{\mathrm{lim}}f(x)=f(a)[/latex].

If a function [latex]f\left(x\right)[/latex] is not continuous at [latex]x=a[/latex], the function is discontinuous at [latex]x=a[/latex] .