A function is, (almost) by definition, continuous at its isolated points.
What is the rationale behind this?
Intuitively it would seem to me that isolated points "shouldn't" be considered points at which the function is continuous. Indeed, often in books/courses, it is noted that somewhat surprisingly, a function is continuous at its isolated points and this is often assigned as an exercise (example).
I am guessing that there are advantages to doing so but I don't know what they are.
Often, we first define $\lim_{x\rightarrow a} f(x)$ if and only if $a$ is a limit point of the domain of $f$.
In which case, in the usual definition of continuity that includes isolated points, we are unable to shorten the definition of continuity to simply: $\lim_{x\rightarrow a} f(x) = f(a)$, because that would fail to include isolated points. So it would seem to me that including isolated points actually complicates the definition of continuity.