Continuity Checker
Check whether a rational function P(x)/Q(x) is continuous at a point — and classify any discontinuity.
Example shown: f(x) = (x² − 4)/(x − 2) at a = 2. Both top and bottom are zero there, but the (x − 2) factor cancels — a removable discontinuity (hole).
① Answer
② Steps
③ Why it works
A function is continuous at a when three things agree: f(a) exists, the two-sided limit exists, and they are equal. Rational functions are continuous at every point of their domain — the only possible trouble is where the denominator is zero. If the numerator is also zero there, a common factor of (x − a) may cancel (a hole); otherwise the function blows up (a vertical asymptote).