What Is Removable Discontinuity?
Removable Discontinuity: A removable discontinuity is a point on the graph that is undefined or does not fit the rest of the graph. There is a gap at that location when you are looking at the graph. When graphed, a removable discontinuity is marked by an open circle on the graph at the point where the graph is undefined or is a different value like this.
There are two ways a removable discontinuity can be created. Let’s talk about the first one now. Do you see it? There is a small open circle at the point where x=2.5 approximately.
A hole in a graph. That is, a discontinuity that can be “repaired” by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can be made connected by filling in a single point.
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Formally, a removable discontinuity is one at which the limit of the function exists but does not equal the value of the function at that point; this may be because the function does not exist at that point.
How do you solve a removable discontinuity?
- Step 1: Factor the numerator and the denominator.
- Step 2: Identify factors that occur in both the numerator and the denominator.
- Step 3: Set the common factors equal to zero.
- Step 4: Solve for x.
What is the difference between a removable and non removable discontinuity?
Non Removable Discontinuity\Removable And Nonremovable Discontinuity
As your pre-calculus teacher will tell you, functions that aren’t continuous at an x value either have a removable discontinuity (a hole in the graph of the function) or a nonremovable discontinuity (such as a jump or an asymptote in the graph):
If the function factors and the bottom term cancels, the discontinuity at the x-value for which the denominator was zero is removable, so the graph has a hole in it.
For example, this function factors as shown:
After canceling, it leaves you with x – 7. Therefore x + 3 = 0 (or x = –3) is a removable discontinuity — the graph has a hole, like you see in Figure a.The graph of removable leaves you feeling empty, whereas a graph of a non-removable discontinuity leaves you feeling jumpy.
If a term doesn’t cancel, the discontinuity at this x value corresponding to this term for which the denominator is zero is nonremovable, and the graph has a vertical asymptote.
The following function factors as shown:
Because the x + 1 cancels, you have a removable at x = –1 (you’d see a hole in the graph there, not an asymptote). But the x – 6 didn’t cancel in the denominator, so you have a nonremovable discontinuity at x = 6. This discontinuity creates a vertical asymptote in the graph at x = 6. Figure b shows the graph of g(x).
What Is A Removable Discontinuity
A removable discontinuity can be created by defining a blip in the graph like this.
The above function tells us that the graph generally follows the function f(x)=x^2-1 except for at the point x=4. When we graph it, we will need to draw a little open circle at the point on the graph and mark that it equals 2 at that point. This is a created discontinuity. If you were the one defining the function, you can easily remove the discontinuity by redefining the function. Looking at the function f(x)=x^2-1, we can calculate that at x=4, f(x)=15. So, if we redefine our point at x=4 to equal 15, we will have removed our
If we were to graph the above, we would get a continuous graph without any discontinuities. When you see functions written out like that, be sure to check whether the function really has a discontinuity or not. Sometimes the function is continuous but just written like it isn’t just to be tricky.
Give An Example Of A Function With Both A Removable And A Non-removable Discontinuity
A real-valued univariate function is said to have a removable discontinuity at a point in its domain provided that both and
exist while . Removable discontinuities are so named because one can “remove” this point of discontinuity by defining an almost everywhere identical function of the form
which necessarily is everywhere-continuous.
The figure above shows the piecewise function
a function for which while . In particular, has a removable discontinuity at due to the fact that defining a function as discussed above and satisfying would yield an everywhere-continuous version of .
Note that the given definition of removable discontinuity fails to apply to functions for which and for which fails to exist; in particular, the above definition allows one only to talk about a function being discontinuous at points for which it is defined. This definition isn’t uniform, however, and as a result, some authors claim that, e.g., has a removable discontinuity at the point . This notion is related to the so-called sinc function.
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