# Removable discontinuity

Here 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 = 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) = 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

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 , 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.