You can remove a removable discontinuity by redefining the function at the “problem spot”. Removable discontinuities are where the graph of a function has a hole. This will occur in rational functions at an x-value that makes both the denominator and the numerator equal zero. For example, the graph of the function defined by
will have a hole at ##x=2## since both the denominator and numerator are zero for this x-value.
We can find the y-value of the hole by substituting 2 for x in the simplified function.
See the image below.
To remove the discontinuity, we will redefine the function at x = 2. This will effectively “fill in the hole.”