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 there is no possible limit of the function at that number. In other words the function fails to be continuous because there is no number $L$ where $\lim_{x \to a} f(x) = L $.

Such examples are commonly used to illustrate functions that are not continuous at a single point because

Consider $f(x) = \sin(\frac 1x)$ when $x \ne 0$ and $f(0)=L$. Then $\lim_{x \to 0} f(x) = \lim_{x \to 0}\sin(\frac 1x) \ne L$ for any value of $L$ so $f$ is not continuous at $x=0$.

In the GeoGebra sketch below with both mapping diagram and graph to visualize the failure of the limit of $f$ at $x=0$.

Martin Flashman Created with GeoGebra

Notice how the points on the graph are paired with the points on the mapping diagram.