How To Remove Removable Discontinuity And Make A Function Continuous At A Point

f has a removable discontinuity at x=a

lim x->a f(x) exists, but does not equal f(a)

A removable discontinuity or a point discontinuity implies that the function f is undefined at a certain point a, but it can be redefined, filled in or extended to make it continuous at a. This can happen if the function is defined piecewise, if it has a hole or if it has a limit at a that is not equal to the function value at a.

To make the function continuous at point a, the function has to be redefined or extended to fill in the hole or remove the jump at a. This can be done by finding the limit of f as x approaches a and then defining f(a) to be the same as the limit.

For example, if f(x)= (x^2-1)/(x-1) has a removable discontinuity at x=1, then we can factor the numerator as (x-1)(x+1) and cancel out the common factor with the denominator (x-1). This leaves f(x)=x+1 for all x≠1. To fill in the hole at x=1, we can define f(1) to be the limit of f as x approaches 1, which is 2. Therefore, the new definition of f is:

f(x)={x+1, x≠1; 2, x=1}

And the function is now continuous at x=1.

It is important to note that removing a discontinuity does not change the behavior of the function on the rest of its domain. It only makes the function continuous at the point a where the discontinuity occurred.

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