CCD.DLC.Example.1. A Removable Discontinuity.
Functions that have a definition that specify a value at one number when
an algebraic rule fails sometimes fail to be continuous at that number
only because the value at that number does not equal the limit of the
function at that number. In other words the function fails to be
continuous because even though there is a number $L$ where
$\lim_{x \to a} f(x) = L $, it turns out that $f(a) \ne L$.
Such examples are commonly used to illustrate functions that are not
continuous
at a single point only because $f(a) \ne L$, whereas if the value of $f$
at $a$ is redefined so that $f(a) = L$, then the discontinuity at $a$
can be removed. Thus these functions are described as having a removable discontinuity at $a$.
Consider $f(x) = \frac{x^2-1}{x-1}$ when $x \ne 1$ and $f(1)=5$.
Then $\lim_{x \to 1} f(x) = \lim_{x \to 1} x+1 = 2 \ne 5=f(1) $
so $f$ is not continuous at $x=1$, but if we consider changing the value
of $f$ so that $f(1)=2$, then $f$ is continuous at $x=1$. So $f$ has a
removable discontinuity at $x=1$.
In the GeoGebra sketch below with
both mapping diagram and graph to visualize the failure of the removable discontinuity of $f$ at $x=1$..
Martin Flashman
Created with GeoGebra
Notice how the points on the graph are paired with the points
on the mapping diagram.