Coupon Accepted Successfully!


Removable discontinuity

Here 88137.png necessarily exists, but is either not equal to f(a) or f(a) is not defined. In this case, therefore it is possible to redefine the function in such a manner that 88131.png = f(a) and thus making the function continuous.
Consider the functions g(x) = (sin x)/x. Function is not defined at x = 0, so domain is R {0}. Since the limit of g at 0 is 1, g can be extended continuously to R by defining its value at 0 to be 1.
Thus redefined function g(x) = 88125.png is continuous at x = 0.
Thus a point in the domain that can be filled in so that the resulting function is continuous is called a removable discontinuity.
Consider function 88119.png
In this example, the function is nicely defined away from the point x = 1.
In fact, if x ≠ 1, the function is
However, if we were to consider the point x = 1, this definition no longer makes sense since we would have to divide by zero. The function instead tells us that the value of the function is f(1) = 3.
In this example, the graph has a “hole” at the point x = 1, which can be filled by redefined f(x) at x = 1 as 2.
This type of discontinuity is also called “missing point discontinuity.”

Non-removable discontinuity

If 88107.png , then f(x) is said to have first kind of non-removable discontinuity.
Consider the functions f(x) = 1/x. Function is not defined at x = 0, The function f cannot be extended to a continuous function whose domain is R, since no matter what value is assigned at 0, the resulting function will not be continuous. A point in the domain that cannot be filled in so that the resulting function is continuous is called a non-removable discontinuity.

Test Your Skills Now!
Take a Quiz now
Reviewer Name