Continuity is something best learned from graphs to get a feel for it. Then you can go to equations to cement the concept in your head. Before we get started, let's watch an introduction video to get an idea of where we are going.
video by Dr Chris Tisdell |
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Recommended Books on Amazon (affiliate links) | ||
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Definition of Continuity
There are 3 parts to continuity. At this moment we are talking about continuity at a point.
For a function f(x) to be continuous at a point x = c all three of these conditions must hold.
1. \( f(c) \) is defined. |
2. \( \displaystyle{\lim_{x \rightarrow c}{f(x)}} \) exists. |
3. \( \displaystyle{\lim_{x \rightarrow c}{f(x)} = f(c)}. \) |
If any one of these conditions is broken, then the function is not continuous at \(x=c\).
Let's look at graphs where each of the above conditions does not hold.
Condition 1: |
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\( f(c) \) is not defined |
This graph shows an example of where the function is not defined at \(x=c\). So this function is not continuous at \(x=c\). This is an example of a removable discontinuity. We can just redefine the function by redefining one point at \(f(c)\) to make it continuous. |
Condition 2: |
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\( \displaystyle{\lim_{x \rightarrow c}{f(x)}} \) does not exist. |
Notice that the limit from the left is different than the limit from the right (at \(x=c\)). This means the limit does not exist. This is an example of a non-removable discontinuity at \(x=c\). There is not way to 'plug the hole' or redefine the function at only one point so that the result is continuous. |
Condition 3: |
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\( \displaystyle{\lim_{x \rightarrow c}{f(x)} \neq f(c)}. \) |
This graph shows an example of where the first two cases hold but the third doesn't, i.e. \(f(c)\) is defined, the limit exists but the limit does not equal \(f(c)\). This is also an example of a removable discontinuity. Notice you can just move the \(f(c)\) to fill the hole to make the function continuous. |
Notes
1. Although not explicitly stated above, continuity holds in both directions, i.e. if a function is continuous then all three conditions hold and if all three conditions hold, then the function is continuous. So we can say, \(f(x)\) is continuous at \(x=c\) if and only if all three conditions listed above hold.
2. For condition 2 above, where the limit does not exist, sometimes we need to look at one-sided limits, i.e. limits from each side of the value we are talking about. You will find discussion, videos and practice problems on the one-sided limits page for this case.
Example Functions - - There are some functions that are guaranteed to be continuous on their domains. This is important . . . these functions are not necessarily continuous everywhere but they are continuous on their domains. We can use this information to build continuity information about other functions.
function type | example |
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polynomials | \(x^3+3x^2+1\) |
rational functions | \( (x+3)/(x^2-1) \) |
root functions | \( \sqrt{x+7} \) |
trig functions | \( \sin(x) \) |
logarithm functions | \( \ln(x-1) \) |
Okay, now that you have an intuitive idea of continuity, let's watch some videos to help you understand and use continuity. It is important to watch both of them to get a complete picture of continuity.
1. This video explains continuity from a more mathematical viewpoint.
video by PatrickJMT |
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2. This next video is a great video to watch to get a much better understanding of continuity using the limit definition. This is one of the best videos we've seen on youtube that explains a complicated math topic in a way that is understandable.
video by PatrickJMT |
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Okay, now that you have a better understanding of continuity, take a look at discontinuities explained next. There is a great video in this section that will help you a lot. The discussion is going to cover several, seemingly diverse, topics. However, they are related in that the resulting equations are similar.
Discontinuities - Removable and Nonremovable
First let's discuss the 3 main types of discontinuities: vertical asymptotes, holes and jumps. Here are three graphs demonstrating each type.
\(\displaystyle{ f(x) = \frac{1}{x-1} }\) |
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1. Vertical Asymptote - - For this graph, at \(x=1\) we have a vertical asymptote. This is a nonremovable discontinuity, which means that we can't redefine the function at \(x=1\) with one value that will make the function continuous there.
Notice in the equation that at \(x=1\), we have a zero in the denominator and a non-zero number in the numerator. The zero in the denominator removes the \(x=1\) from the domain and the non-zero number in the numerator tells us that we have an asymptote, consequently, a non-removable discontinuity.
\(\displaystyle{ g(x)=\frac{x^2-1}{x-1} }\) |
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2. Hole - - For this graph, at \(x=1\) we have a hole. This is a removable discontinuity since if we add the single point \((1,2)\) to the function, the result is a continuous function at \(x=1\).
Notice in the equation of \(g(x)\) that at \(x=1\), we have zero in the numerator and the denominator.
3. Jump - - In this third plot at \(x=1\) we have a jump. This is a nonremovable discontinuity since we can't redefine the function a single point to make it continuous at \(x=1\).
Note: This graph is a piecewise function. For a review, go to the precalculus piecewise function page.
\(\displaystyle{ h(x) = \left\{ \begin{array}{rcl} x+1 & & x \leq 1 \\ x^2 & & x > 1 \end{array} \right. }\) |
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Here are a couple of great videos explaining discontinuities in more detail with lots of examples. Her example in the second video of what she calls a crazy graph is especially good. As an instructor, I have often put questions like this on exams.
video by Krista King Math |
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video by Krista King Math |
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Zeroes, Holes and Asymptotes
Now let's look at the equations for each of these. I have included this with the discussion of discontinuities since two of these are discontinuities and the third, zeroes, are related to the other two but are not discontinuities. For this discussion, we are going to look how the equations are similar.
First let's look at zeroes. Zeroes of a function are sometimes called poles (especially in electrical engineering). Basically, they are points where the graph of a function crosses the x-axis, i.e. where \(y=0\). They are not discontinuities but are important points in mathematics and engineering. If you have a function that is a fraction such as \(\displaystyle{ f(x)=\frac{n(x)}{d(x)} }\), zeroes occur at x-values where the numerator function is zero but the denominator function is NOT zero. For example, look at the second graph above. A zero occurs at \(x = -1\). You can also say that there is a zero at the point \((-1,0)\). By definition, the y-value is zero, so we usually do not write the point \((-1,0)\). We usually just say \(x=-1\) or at \(-1\).
Okay, let's look at holes. If we have the a function in fraction form that looks like \(\displaystyle{ f(x)=\frac{n(x)}{d(x)} }\), holes occur at x-values where the numerator AND denominator are both zero at the same x-value. A hole is a discontinuity. Looking at the second graph above, we have a hole at \(x=1\) because the numerator and denominator of g(x) are both zero at \(x=1\).
Finally, vertical asymptotes occur at x-values where the denominator is zero but the numerator is NOT zero. Asymptotes are discontinuities. The first graph above shows this case. Notice that at \(x=1\), the numerator of f(x) is 1, but the denominator is zero.
Let's sum this up. For a fractional function in the form \(\displaystyle{ f(x)=\frac{n(x)}{d(x)} }\):
- Zeroes occur at x-values where \(n(x) = 0\) and \(d(x) \neq 0\)
- Holes occur at x-values where \(n(x) = 0\) and \(d(x) = 0\)
- Vertical Asymptotes occur at x-values where \(n(x) \neq 0\) and \(d(x) = 0\)
Here is the same information in table form.
\(f(x)=n(x)/d(x)\) | |||
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\(n(x) \neq 0\) |
\(n(x) = 0\) |
Discontinuity? | |
\(d(x) \neq 0\) | Zero | No | |
\(d(x) = 0\) | Vertical Asymptote | Hole | Yes |
Practice
\(\displaystyle{ g(x) = \left\{ \begin{array}{lr} x-3 & x \leq -1 \\ x^2+1 & -1 < x \leq 2 \\ x^3+4 & x >2 \end{array} \right. }\)
At what points is \(g(x)\) NOT continuous?
Problem Statement
\(\displaystyle{ g(x) = \left\{ \begin{array}{lr} x-3 & x \leq -1 \\ x^2+1 & -1 < x \leq 2 \\ x^3+4 & x >2 \end{array} \right. }\)
At what points is \(g(x)\) NOT continuous?
Solution
video by PatrickJMT |
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\(\displaystyle{ f(x) = \left\{ \begin{array}{lr} x^2+3x & x < 0 \\ \sqrt{x}+1 & x \geq 0 \end{array} \right. }\)
At what points is \(f(x)\) NOT continuous?
Problem Statement
\(\displaystyle{ f(x) = \left\{ \begin{array}{lr} x^2+3x & x < 0 \\ \sqrt{x}+1 & x \geq 0 \end{array} \right. }\)
At what points is \(f(x)\) NOT continuous?
Solution
video by PatrickJMT |
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\(\displaystyle{ f(x) = \left\{ \begin{array}{lr} 2x+1 & x \leq -1 \\ 3x & -1 < x < 1 \\ 2x-1 & x \geq 1 \end{array} \right. }\)
At what points is \(f(x)\) NOT continuous?
Problem Statement
\(\displaystyle{ f(x) = \left\{ \begin{array}{lr} 2x+1 & x \leq -1 \\ 3x & -1 < x < 1 \\ 2x-1 & x \geq 1 \end{array} \right. }\)
At what points is \(f(x)\) NOT continuous?
Solution
video by PatrickJMT |
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Find where \(\displaystyle{f(x)=\sqrt[3]{\frac{x+1}{x-1}}}\) is continuous.
Problem Statement
Find where \(\displaystyle{f(x)=\sqrt[3]{\frac{x+1}{x-1}}}\) is continuous.
Solution
video by Krista King Math |
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Classify any discontinuities of the function \(\displaystyle{ f(x)=\frac{x-2}{x^2-4} }\), then redefine the function at any removable discontinuities to make it continuous.
Problem Statement
Classify any discontinuities of the function \(\displaystyle{ f(x)=\frac{x-2}{x^2-4} }\), then redefine the function at any removable discontinuities to make it continuous.
Solution
video by Krista King Math |
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\(\displaystyle{ f(x) = \left\{ \begin{array}{lr} x^2+3 & x < 1 \\ ax+6 & x \geq 1 \end{array} \right. }\)
What value of a makes the piecewise function \(f(x)\) continuous on the entire real line?
Problem Statement
\(\displaystyle{ f(x) = \left\{ \begin{array}{lr} x^2+3 & x < 1 \\ ax+6 & x \geq 1 \end{array} \right. }\)
What value of a makes the piecewise function \(f(x)\) continuous on the entire real line?
Solution
video by PatrickJMT |
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\(\displaystyle{ f(x) = \left\{ \begin{array}{lr} c^2 - x^2 & x < 0 \\ 2(x-c)^2 & x \geq 0 \end{array} \right. }\)
What value of c makes the piecewise function \(f(x)\) continuous on the entire real line?
Problem Statement
\(\displaystyle{ f(x) = \left\{ \begin{array}{lr} c^2 - x^2 & x < 0 \\ 2(x-c)^2 & x \geq 0 \end{array} \right. }\)
What value of c makes the piecewise function \(f(x)\) continuous on the entire real line?
Solution
video by Krista King Math |
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