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What Is Removable Discontinuity?

What Is Removable Discontinuity?

Removable Discontinuity: 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.

What Is Removable Discontinuity?

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.

Removable Discontinuities, cont.
  1. Step 1: Factor the numerator and the denominator.
  2. Step 2: Identify factors that occur in both the numerator and the denominator.
  3. Step 3: Set the common factors equal to zero.
  1. Step 4: Solve for x.
[Calculus 1] What is the difference between a removable and non removable discontinuity? … If the limit does not exist, then the discontinuity is nonremovableIn essence, if adjusting the function’s value solely at the point of discontinuity will render the function continuous, then the discontinuity is removable.


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 a removable discontinuity leaves you feeling empty, whereas a graph of a nonremovable

    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.

Defining a blip.
removable discontinuity

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

removable discontinuity

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 f=f(x) is said to have a removable discontinuity at a point x_0 in its domain provided that both f(x_0) and


exist while f(x_0)!=L. Removable discontinuities are so named because one can “remove” this point of discontinuity by defining an almost everywhere identical function F=F(x) of the form

 F(x)={f(x) for x!=x_0; L for x=x_0,

which necessarily is everywhere-continuous.


The figure above shows the piecewise function

 f(x)={(x^2-1)/(x-1) for x!=1; 5/2 for x=1,

a function for which lim_(x->1-)f(x)=lim_(x->1+)f(x)=2 while f(1)=5/2. In particular, f has a removable discontinuity at x=1 due to the fact that defining a function F(x) as discussed above and satisfying F(1)=2 would yield an everywhere-continuous version of f.

Note that the given definition of removable discontinuity fails to apply to functions f for which lim_(x->x_0)f(x)=L and for which f(x_0) 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., f(x)=sin(x)/x has a removable discontinuity at the point x=0. This notion is related to the so-called sinc function.

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