Math Lesson 16.5.3 - Continuity of a Piecewise Function

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Welcome to our Math lesson on Continuity of a Piecewise Function, this is the third lesson of our suite of math lessons covering the topic of Piecewise Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Continuity of a Piecewise Function

Some functions are undefined for certain values of the independent variable x. For example, the reciprocal function f(x) = 1/x is not defined for x = 0 as the division by 0 is not allowed in the set of real numbers. We have seen that the graph of such a function contains two different parts that have no common points, i.e. they are not connected with each other.

Basically, there are two types of functions in regard to their graph: continuous and non-continuous. By definition, a function is continuous if its graph has no interruptions. In other words, if you can plot a graph without taking your hand off the sheet, then you are dealing with the graph of a continuous function. If you have no other choice but to sketch the graph in two or more steps by taking off the hand from the sheet, then you are dealing with a non-continuous function, where both the domain and range are obtained by considering two or more separate intervals in the corresponding axes. Look at the figure below where the graph of the piecewise function

is shown.

You can easily see that the two parts of the graph have no common points. This means this function is not continuous.

On the other hand, if we consider the piecewise function

is continuous because the two graphs converge at a common point (at x = 2). Look at the graph below.

How to know whether a Piecewise Function is Continuous or Not without Plotting the Graph?

You can check the continuity of a piecewise function by finding its value at the boundary (limit) point x = a. If the two pieces give the same output for this value of x, then the function is continuous. Let's explain this point through an example.

Example 3

Check the continuity of the following piecewise functions without plotting the graph.

1. f(x) = 2x + 1 x ≥ 14 - x x < 1
2. f(x) = 2 - 3x x ≥ 0x² - 2x + 2 x > 0
3. f(x) = 1 - x² x ≥ -13 - 5x x < -1

Solution 3

1. We must check the continuity of this function at x = -1. If the value of the two pieces at this point is equal, the function is continuous. Thus, for the top part of f(x) we have
   f(x) = 2x + 1
   f(x) = 2 ∙ 1 + 1
   = 2 + 1
   = 3
   and for the bottom part of f(x) we have
   f(x) = 4 - x
   f(x) = 4 - 1
   = 3
   Thus, since the two values of f(x) are equal, the function is continuous at x = 1.
2. We must check the continuity of this function at x = 0. If the value of the two pieces at this point is equal, the function is continuous. Thus, for the top part of f(x) we have
   f(x) = 2 - 3x
   f(0) = 2 - 3 ∙ 0
   = 2 - 0
   = 2
   and for the bottom part of f(x) we have
   f(x) = x² - 2x + 2
   f(x) = 0² - 2 · 0 + 2
   = 0 - 0 + 2
   = 2
   Thus, since the two values of f(x) are equal, the function is continuous at x = 1.
3. We must check the continuity of this function at x = -1. If the value of the two pieces at this point is equal, the function is continuous. Thus, for the top part of f(x) we have
   f(x) = 1-x²
   f(-1) = 1 - (-1)²
   = 1 - 1
   = 0
   and for the bottom part of f(x) we have
   f(x) = 3 - 5x
   f(-1) = 3 - 5 ∙ (-1)
   = 3 + 5
   = 8
   Thus, since the two values of f(x) are different at x = -1, the function is non-continuous at this point.

You have reached the end of Math lesson 16.5.3 Continuity of a Piecewise Function. There are 6 lessons in this physics tutorial covering Piecewise Functions, you can access all the lessons from this tutorial below.

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