# 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

**= 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***x***(1) = 3.***f*In this example, the graph has a

**“hole”**at the point*x*= 1, which can be filled by redefined**(***f***) at***x***= 1 as 2.***x*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**, 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.***R*