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Continuous Functions

Author: Sophia
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what's covered
In this lesson, you will learn what it means for a function to be continuous, including how limits are used in relation to continuity. Specifically, this lesson will cover:

Table of Contents

1. The Definition of Continuity

A function is called continuous at a point where there is no break in the graph at that point.

That is, limit as x rightwards arrow a of f open parentheses x close parentheses equals f open parentheses a close parentheses.

Consider the graph of y equals f open parentheses x close parentheses shown below. We will examine the continuity of f open parentheses x close parentheses when x = 1, 2, 3, and 4.

A graph with an x-axis ranging from 0 to 4 and a y-axis ranging from –1 to 2. The graph represents the function y equals f(x) and has two curves above the x-axis and a closed dot at (2, 1). The first curve rises from the left of the y-axis toward the first quadrant by crossing the y-axis at (0, 1.8), peaks slightly, and then descends to a closed dot at (1, 2). From here, the curve dips slightly and rises again to connect to an open circle at (2, 2), then rises to (2.5, 2.3), then descends to a closed dot at (3, 2). The second curve rises from an open circle at (3, 1) to a peak at an open circle at (4, 2), and then starts descending.

Given Point Continuity of bold italic f open parentheses bold x close parentheses at the Given Point
x equals 1 The graph of f open parentheses x close parentheses is continuous when x equals 1 since there are no breaks in the graph at that point. Looking just before x equals 1 comma the graph passes through the point open parentheses 1 comma space f open parentheses 1 close parentheses close parentheses and continues to “flow” afterwards.
x equals 2 The graph of f open parentheses x close parentheses is NOT continuous when x equals 2. There is a hole in the graph when x equals 2 comma meaning there is a break in the graph.
x equals 3 The graph is NOT continuous when x equals 3. There is a break in the graph.
x equals 4 The graph is NOT continuous when x equals 4. There is a hole in the graph.

Now, considering these 4 points, let’s examine the limits at these points and the values of f open parentheses x close parentheses at these points as well as whether or not the function is continuous at these points:

x-value stack bold l bold i bold m with bold x bold rightwards arrow bold a below bold italic f open parentheses bold x close parentheses bold italic f open parentheses bold a close parentheses Continuous?
x equals 1 limit as x rightwards arrow 1 of f open parentheses x close parentheses equals 2 f open parentheses 1 close parentheses equals 2 Yes
x equals 2 limit as x rightwards arrow 2 of f open parentheses x close parentheses equals 2 f open parentheses 2 close parentheses equals 1 No
x equals 3 limit as x rightwards arrow 3 of f open parentheses x close parentheses does not exist (the left-hand and right-hand limits are not equal). f open parentheses 3 close parentheses equals 2 No
x equals 4 limit as x rightwards arrow 4 of f open parentheses x close parentheses equals 2 f open parentheses 4 close parentheses is not defined. No

From this table, we can conclude the following:

  • A function f open parentheses x close parentheses is continuous at x equals a if limit as x rightwards arrow a of f open parentheses x close parentheses equals f open parentheses a close parentheses. That is, limit as x rightwards arrow a of f open parentheses x close parentheses exists and is equal to the value of f open parentheses a close parentheses.
  • A function f open parentheses x close parentheses is not continuous at x equals a if any of the following occur:
    • limit as x rightwards arrow a of f open parentheses x close parentheses does not exist.
    • f open parentheses a close parentheses is undefined.
    • limit as x rightwards arrow a of f open parentheses x close parentheses exists, but is not equal to f open parentheses a close parentheses.
term to know
Continuous Function
A function that has no breaks in the graph. That is, limit as x rightwards arrow a of f open parentheses x close parentheses equals f open parentheses a close parentheses.

2. Determining if a Function Is Continuous at x = a

To determine if a function is continuous at x equals a comma we need to compare the values of limit as x rightwards arrow a of f open parentheses x close parentheses and f open parentheses a close parentheses. While computing f open parentheses a close parentheses is straightforward, computing limit as x rightwards arrow a of f open parentheses x close parentheses requires more care, and sometimes requires one-sided limits.

EXAMPLE

Consider the function f open parentheses x close parentheses equals open curly brackets table attributes columnalign left center end attributes row cell 2 x plus 1 end cell cell i f space x less than 4 end cell row cell open parentheses x minus 1 close parentheses squared end cell cell i f space x greater or equal than 4 end cell end table close. Determine if f open parentheses x close parentheses is continuous at x equals 4.

First, check to see if limit as x rightwards arrow 4 of f open parentheses x close parentheses exists. Since f open parentheses x close parentheses changes definition when x equals 4 comma we need to consider the one-sided limits:
  • Left-sided limit: limit as x rightwards arrow 4 to the power of short dash of f open parentheses x close parentheses equals limit as x rightwards arrow 4 to the power of short dash of open parentheses 2 x plus 1 close parentheses equals 2 open parentheses 4 close parentheses plus 1 equals 9
  • Right-sided limit: limit as x rightwards arrow 4 to the power of plus of f open parentheses x close parentheses equals limit as x rightwards arrow 4 to the power of plus of open parentheses x minus 1 close parentheses squared equals open parentheses 4 minus 1 close parentheses squared equals 9
  • Conclusion: limit as x rightwards arrow 4 of f open parentheses x close parentheses equals 9 comma which means it exists and is equal to 9.
From looking at the function definition, f open parentheses 4 close parentheses equals open parentheses 4 minus 1 close parentheses squared equals 9. Thus, the limit and the function value are the same; therefore the function is continuous at x equals 4. Here is the graph of f open parentheses x close parentheses to help visualize this:

A graph with an x-axis ranging from –6 to 6 and a y-axis ranging from –1 to 11. A line slants upward from the third quadrant to the first quadrant by crossing the x-axis between –1 and 0 and the y-axis at –1 and reaches a closed circle at (4, 9). The line then bends slightly toward the y-axis and extends further.

EXAMPLE

Consider the function f open parentheses x close parentheses equals open curly brackets table attributes columnalign left center end attributes row cell fraction numerator x squared minus x minus 12 over denominator x squared minus 16 end fraction end cell cell i f space x not equal to 4 end cell row 5 cell i f space x equals 4 end cell end table close. Determine if f open parentheses x close parentheses is continuous at x equals 4.

First, evaluate limit as x rightwards arrow 4 of f open parentheses x close parentheses. Since f open parentheses x close parentheses equals open curly brackets table attributes columnalign left center end attributes row cell fraction numerator x squared minus x minus 12 over denominator x squared minus 16 end fraction end cell cell i f space x not equal to 4 end cell row 5 cell i f space x equals 4 end cell end table close is defined on both sides of x equals 4 comma there is no need to compute one-sided limits.

To evaluate the limit, first consider direct substitution:

limit as x rightwards arrow 4 of fraction numerator x squared minus x minus 12 over denominator x squared minus 16 end fraction equals fraction numerator 4 squared minus 4 minus 12 over denominator 4 squared minus 16 end fraction equals 0 over 0

While this is undefined, notice that both the numerator and denominator are 0. This means that the expression can be manipulated in order to evaluate the limit.

limit as x rightwards arrow 4 of fraction numerator x squared minus x minus 12 over denominator x squared minus 16 end fraction equals limit as x rightwards arrow 4 of fraction numerator open parentheses x minus 4 close parentheses open parentheses x plus 3 close parentheses over denominator open parentheses x plus 4 close parentheses open parentheses x minus 4 close parentheses end fraction Factor the expression.
equals limit as x rightwards arrow 4 of fraction numerator x plus 3 over denominator x plus 4 end fraction Remove the common factor of x minus 4.
equals fraction numerator 4 plus 3 over denominator 4 plus 4 end fraction Direct substitution works since the denominator is not 0.
equals 7 over 8 Simplify.

However, f open parentheses 4 close parentheses equals 5. Since the limit and the function value are different, this function is not continuous at x equals 4. Here is a graph to help visualize this:

A graph with an x-axis ranging from –6 to 6 and a y-axis ranging from –5 to 7. The graph has two curves and a closed dot at (4, 5). One curve initially rises gently from (–7, 1.2) to (–5, 2) and then rises vertically upward in the second quadrant, getting closer and closer to x equals -4. The second curve rises steeply from the third quadrant just to the right of x equals -4. crossing into the second quadrant at (-3, 0), then begins to level out toward y equals 1, passing through the y-axis at the point (0, 0.75). The curve also contains an open circle at the point (4, 0.875).

try it
Consider the function: f open parentheses x close parentheses equals open curly brackets table attributes columnalign left center end attributes row cell 3 x plus 4 end cell cell i f space x less than 1 end cell row cell square root of x plus 8 end root end cell cell i f space x greater or equal than 1 end cell end table close
Determine if f   (x  ) is continuous when x = 1.
Find the limit as x rightwards arrow 1 from each side:

limit as x rightwards arrow 1 to the power of minus of f open parentheses x close parentheses equals limit as x rightwards arrow 1 to the power of minus of open parentheses 3 x plus 4 close parentheses equals 3 open parentheses 1 close parentheses plus 4 equals 7

limit as x rightwards arrow 1 to the power of plus of f open parentheses x close parentheses equals limit as x rightwards arrow 1 to the power of plus of square root of x plus 8 end root equals square root of 1 plus 8 end root equals square root of 9 equals 3

Since the one-sided limits are not equal, f open parentheses x close parentheses is not continuous at x equals 1.

3. Determining Intervals Over Which a Function Is Continuous

For a function to be continuous on an interval of values, it has to be continuous at every point contained in the interval.

EXAMPLE

f open parentheses x close parentheses equals x squared minus 4 x plus 5 is continuous at every real number. Thus, we say that f open parentheses x close parentheses is continuous on the interval open parentheses short dash infinity comma space infinity close parentheses.

EXAMPLE

f open parentheses x close parentheses equals fraction numerator 2 over denominator x minus 1 end fraction is continuous at every value except x equals 1. We can say that f open parentheses x close parentheses is continuous on the intervals open parentheses short dash infinity comma space 1 close parentheses and open parentheses 1 comma space infinity close parentheses. This can also be written as open parentheses short dash infinity comma space 1 close parentheses union open parentheses 1 comma space infinity close parentheses.

It is also possible to define continuity at an endpoint. For example, consider f open parentheses x close parentheses equals square root of 6 x minus x squared end root comma whose graph is shown below. Note that the domain of this function is open square brackets 0 comma space 6 close square brackets.

A graph with an x-axis and a y-axis ranging from –6 to 6. A semicircle starts from a marked point at (0, 0) and ends at another marked point at (6, 0) by passing through a marked point at (3, 3) in the first quadrant.

This means that defining continuity at x equals 0 and x equals 6 takes a bit more care.

Consider the endpoint x equals 0. It can only be approached from the right. Looking at the graph, observe that limit as x rightwards arrow 0 to the power of plus of f open parentheses x close parentheses equals 0 and f open parentheses 0 close parentheses equals 0.

Consider the endpoint x equals 6. It can only be approached from the left. Looking at the graph, observe that limit as x rightwards arrow 6 to the power of short dash of f open parentheses x close parentheses equals 0 and f open parentheses 6 close parentheses equals 0.

big idea
A function is continuous from the left at x equals a if limit as x rightwards arrow a to the power of short dash of f open parentheses x close parentheses equals f open parentheses a close parentheses.
A function is continuous from the right at x equals a if limit as x rightwards arrow a to the power of plus of f open parentheses x close parentheses equals f open parentheses a close parentheses.

Thus, in the previous problem, we can say that f open parentheses x close parentheses is continuous from the left at x equals 6 and continuous from the right at x equals 0. This enables us to say that f open parentheses x close parentheses is continuous for all values on the interval open square brackets 0 comma space 6 close square brackets.

EXAMPLE

Determine the interval(s) over which f open parentheses x close parentheses equals square root of x minus 4 end root is continuous. The graph is shown below.

A graph with an x-axis ranging from 2 to 16 and a y-axis. A curve resembling half a sideways parabola starts from a closed dot at (4, 0) on the x-axis and rises gently in the first quadrant.

Note that the domain of f open parentheses x close parentheses is left square bracket 4 comma space infinity right parenthesis. It follows that f open parentheses x close parentheses is continuous on the interval left square bracket 4 comma space infinity right parenthesis comma noting that it is continuous from the right at x equals 4.

try it
Consider the following table:
Function Continuous Interval
f open parentheses x close parentheses equals 3 x minus x to the power of 4 ?
g open parentheses x close parentheses equals fraction numerator x over denominator x plus 4 end fraction ?
h open parentheses x close parentheses equals square root of 2 x minus 1 end root ?

Determine the interval(s) over which each function is continuous.
Function Continuous Interval
f open parentheses x close parentheses equals 3 x minus x to the power of 4 open parentheses short dash infinity comma space infinity close parentheses
g open parentheses x close parentheses equals fraction numerator x over denominator x plus 4 end fraction open parentheses short dash infinity comma space short dash 4 close parentheses union open parentheses short dash 4 comma space infinity close parentheses
h open parentheses x close parentheses equals square root of 2 x minus 1 end root open square brackets 1 half comma space infinity close parentheses

terms to know
Continuous From the Left
A function is continuous from the left at x equals a if limit as x rightwards arrow a to the power of short dash of f open parentheses x close parentheses equals f open parentheses a close parentheses.
Continuous From the Right
A function is continuous from the right at x equals a if limit as x rightwards arrow a to the power of plus of f open parentheses x close parentheses equals f open parentheses a close parentheses.

summary
In this lesson, you learned the definition of continuity, understanding that when given a graph, continuity is determined by locations where the graph has no breaks, jumps, or holes. A continuous function has no breaks in the graph; that is, limit as x rightwards arrow a of f open parentheses x close parentheses equals f open parentheses a close parentheses. You learned that you can use limits to determine if a function is continuous at x equals a (a specific point) by comparing the values of limit as x rightwards arrow a of f open parentheses x close parentheses and f open parentheses a close parentheses. It's important to note that while computing f open parentheses a close parentheses is straightforward, computing limit as x rightwards arrow a of f open parentheses x close parentheses requires more care, and sometimes requires one-sided limits. Lastly, you learned that by examining the domain of a function, you can use it to determine the intervals over which a function is continuous, noting that the function has to be continuous at every point contained in the interval in order to say the function is continuous on the interval.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM CHAPTER 1 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES.

Terms to Know
Continuous From the Left

A function is continuous from the left at x = a if limit as x rightwards arrow a to the power of short dash of f open parentheses x close parentheses equals f open parentheses a close parentheses.

Continuous From the Right

A function is continuous from the left at x = a if limit as x rightwards arrow a to the power of plus of f open parentheses x close parentheses equals f open parentheses a close parentheses.

Continuous Function

A function that has no breaks in the graph. That is, limit as x rightwards arrow a of f open parentheses x close parentheses equals f open parentheses a close parentheses.

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