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Discrete and Continuous Functions

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Continuity and Discontinuity

Continuity is a property of functions that can be drawn without lifting your pencil. Some functions, like the reciprocal functions, have two distinct parts that are unconnected. Functions that are unconnected are discontinuous. What are the three ways functions can be discontinuous and how do they come about? 

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http://www.youtube.com/watch?v=sNWTxomEMEE

Guidance

Functions that can be drawn without lifting up your pencil are called continuous functions. You will define continuous in a more mathematically rigorous way after you study limits. 

There are three types of discontinuities: Removable, Jump and Infinite. 

Removable discontinuities occur when a rational function has a factor with an  x that exists in both the numerator and the denominator. Removable discontinuities are shown in a graph by a hollow circle that is also known as a hole. Below is the graph for f(x) = \frac{(x + 2) (x + 1)}{x + 1} . Notice that it looks just like y = x + 2  except for the hole at x = -1 . There is a hole at x = -1  because when  x = -1, f(x) = \frac{0}{0} .

Removable discontinuities can be “filled in” if you make the function a piecewise function and define a part of the function at the point where the hole is. In the example above, to make f(x)  continuous you could redefine it as:

f(x)=\begin{cases} \frac{(x + 2) (x + 1)}{x + 1},  & x \neq -1 \\1, & x = -1 \end{cases}

Jump discontinuities occur when a function has two ends that don’t meet even if the hole is filled in. In order to satisfy the vertical line test and make sure the graph is truly that of a function, only one of the end points may be filled. Below is an example of a function with a jump discontinuity.

Infinite discontinuities occur when a function has a vertical asymptote on one or both sides. This is shown in the graph of the function below at x = 1 .

Example A

Identify the discontinuity of the function algebraically and then graph the function.

f(x) = \frac{(x - 2) (x + 2) (x - 1)}{(x - 1)}

Solution: Since the factor x - 1  is in both the numerator and the denominator, there is a removable discontinuity at x = 1 .

When graphing the function, you should cancel the removable factor, graph like usual and then insert a hole in the appropriate spot at the end.

Example B

Identify the discontinuity of the piecewise function graphically. 

f(x) =\begin{cases} x^2 - 4  & x < 1\\-1 & x = 1\\- \frac{1}{2} x + 1 & x > 1 \end{cases}

Solution: There is a jump discontinuity at x = 1 . The piecewise function describes a function in three parts; a parabola on the left, a single point in the middle and a line on the right.

Example C

Identify the discontinuity of the function below.

Solution: Since there is a vertical asymptote at x = 1 , this is an infinite discontinuity. 

Concept Problem Revisited

There are three ways that functions can be discontinuous. When a rational function has a vertical asymptote as a result of the denominator being equal to zero at some point, it will have an infinite discontinuity at that point. When the numerator and denominator of a rational function have one or more of the same factors, there will be removable discontinuities corresponding to each of these factors. Finally, when the different parts of a piecewise function don’t “match”, there will be a jump discontinuity.

Vocabulary

Removable discontinuities are also known as holes.  They occur when factors can be algebraically canceled from rational functions. 

Jump discontinuities occur most often with piecewise functions when the pieces don’t match up. 

Infinite discontinuities occur when a factor in the denominator of the function is zero. 

Guided Practice

1. Describe the continuity or discontinuity of the function f(x) = \sin \left( \frac{1}{x} \right) .

2. Describe the discontinuities of the function below.

3. Describe the discontinuities of the function below.

Answers:

1. The function seems to oscillate infinitely as  x approaches zero. One thing that the graph fails to show is that 0 is clearly not in the domain. The graph does not shoot to infinity, nor does it have a simple hole or jump discontinuity. Calculus and Real Analysis are required to state more precisely what is going on. 

2. There is a jump discontinuity at x = -1  and an infinite discontinuity at x = 2 .

3. There are jump discontinuities at x = -2  and x = 4 . There is a removable discontinuity at x = 2 . There is an infinite discontinuity at x = 0 .

Practice

Describe any discontinuities in the functions below:

1.  y = x

2.  y = x^2

3. y = x^3

4.  y = \sqrt{x}

5.  y = \frac{1}{x}

6.  y = e^x

7.  y = \ln (x)

8.  y = \frac{1}{1 + e^{-x}}

9. 

10.

11.

12.  f(x) has a jump discontinuity at x = 3 , a removable discontinuity at x = 5 , and another jump discontinuity at x = 6 . Draw a picture of a graph that could be f(x) .

13.  g(x) has a jump discontinuity at x = -2 , an infinite discontinuity at x = 1 , and another jump discontinuity at x = 3 . Draw a picture of a graph that could be g(x) .

14.  h(x) has a removable discontinuity at x = -4 , a jump discontinuity at x = 1 , and another jump discontinuity at x = 7 . Draw a picture of a graph that could be h(x) .

15. j(x)  has an infinite discontinuity at x = 0 , a removable discontinuity at x = 1 , and a jump discontinuity at x = 4 . Draw a picture of a graph that could be j(x) .

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