If we define a continuous function as a function that is continuous at all points in its domain, then would a removable discontinuity not impact this?
For example consider the function $f(x) = x$. It is clearly continuous at every point in its domain, which is all real numbers.
But now let's have $f(x) = x \cdot \frac{x-2}{x-2}$ which means the function is undefined at $x=2$, but still equivalent to $f(x)=x$ everywhere else, so we exclude that point $x=2$ from the domain.
The domain is now something like $(-\infty, 2) \cup (2, \infty)$. Continuity at a point usually means the limits exist and they equal the function's value at that point. The number $2$ is not in the domain so we only have to consider numbers to the left and to the right of $2$, and no matter what number we pick, they will be defined and have limits equaling those values.
So would we still call such a function "continuous" despite having this $f(2)$ undefined?
Best Answer
Yes, of course. The only thing that matters for continuity are those points at which the function is defined. What occurs outside the domain is irrelevant.