Mathematics

A removable discontinuity at $x=2$ means the limit exists as you approach that point, but the function either isn't defined there or has a different value than the limit—so the definition of continuity, which requires $\lim_{x \to 2} f(x) = f(2)$, is violated. Even though the "hole" could theoretically be filled by redefining $f(2)$, right now the function fails the continuity test because that equality doesn't hold.

A removable discontinuity at $x=2$ means the limit exists as you approach that point, but the function either isn't defined there or has a different value than the limit—so the definition of continuity, which requires $\lim_{x \to 2} f(x) = f(2)$, is violated. Even though the "hole" could theoretically be filled by redefining $f(2)$, right now the function fails the continuity test because that equality doesn't hold.