A
point of discontinuity of a is a with the
x- that makes the function value . A point of discontinuity can also be considered as an of the .
Removable Discontinuity
If a function can be redefined so that its point of discontinuity is removed, it is called a
removable discontinuity. A removable discontinuity may occur when there is a with common in the and . Consider the following .
f(x)=x−2x2−4
For this function,
x=2 is a point of discontinuity because its denominator is
0 when
x=2. Next, notice that the function can be simplified using a .
f(x)=x−2x2−4
f(x)=x−2x2−22
f(x)=x−2(x+2)(x−2)
f(x)=x+2,x=2
After simplifying, the graph behaves as a except in the point of discontinuity, where its value is undefined.
However, the function can be made continuous by redefining it at
x=2. If
2 is substituted for
x in the function
f(x)=x+2, it is found that
f(2)=4. Therefore, the following function is continuous at
x=2.
g(x)=⎩⎪⎪⎨⎪⎪⎧x−2x2−4,4,if x=2if x=2
This means that
x=2 is a removable discontinuity. Note that a removable discontinuity is typically a
hole
in the graph.
Non-Removable Discontinuity
When a function cannot be redefined so that the point of discontinuity becomes a valid , it is called a non-removable discontinuity. Consider, for example, y=x+11. Since x=-1 is not in the of the function, there is a point of discontinuity at x=-1.
This is a non-removable discontinuity because there is no way to redefine the function so that it becomes continuous at x=-1.