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You may have already heard of a continuous function, which is a function on a graph that is a continuous curve. When you put your pencil down to draw it, you never lift your pencil up until the function is complete.
A discontinuous function is the opposite. It is a function that is not a continuous curve, meaning that it has points that are isolated from each other on a graph. When you put your pencil down to draw a discontinuous function, you must lift your pencil up at least one point before it is complete.
If you ever see a function with a break of any kind in it, then you know that function is discontinuous. In the function we have here, you can see how the function keeps going with a break.
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The discontinuous function stops where x equals 1 and y equals 2, and picks up again where x equals 1 and y equals 4.
There are some properties that are specific to discontinuous functions, and two are particularly important:
First, the function always breaks off at a certain point or multiple points. As we have already discussed, discontinuous functions have points where the graph just stops and picks up somewhere else.
Second, the limit of the function at a point of discontinuity is undefined for most discontinuous functions, but not in all cases. The limit can be defined but is still considered discontinuous.
Now, let's explore some of the common types of discontinuous functions.
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One type of discontinuity is called a removable discontinuity, or a hole. It is called removable because the point can be redefined to make the function continuous by matching the value at that point with the rest of the function.
When graphed, a removable discontinuity, or a hole, is just a missing value in the function. Everything else looks like a continuous graph. If we define that missing point, we will have removed the discontinuity.
The removable discontinuity is noted on the graph by a little circle at the point of discontinuity. Do you see how if we define that particular point to be the same as the function at that point, we will have removed the discontinuity?
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Another kind of discontinuity is called a jump discontinuity. This type of discontinuity is where the graph stops at a point and picks up at a completely different point.
If we take the limit of the function as it approaches the point from either side, we will get a different answer. Looking at the graph, if we approach the discontinuity from the left hand side, it looks like we arrive at 2. If we approach the discontinuity from the right hand side, we arrive at what looks like 4. That's why we call this type of function a jump discontinuity; it jumps from one value to another at a certain point.
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In an asymptotic discontinuity, the graphs approaches a point but never touches it,. The graph goes towards infinity at certain points. As you can see in our graph, it goes either all the way up or all the way down towards either positive or negative infinity. The graph avoids the asymptote. In the case of our graph, the asymptote is x equals 0.
Discontinuous functions are functions that are not a continuous curve - there is a hole or jump in the graph. It is an area where the graph cannot continue without being transported somewhere else.
There are many types of continuities. In a removable discontinuity, the point can be redefined to make the function continuous by matching the value at that point with the rest of the function. In a jump discontinuity, the graph stops at a point and picks up at a completely different point. In an asymptotic discontinuity, the graph approaches a point but never touches it, and the graph goes towards infinity at certain points.
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Precalculus: Homework Help Resource11 chapters | 88 lessons
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