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

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.