A function is continuous if you can draw its graph without lifting your pen — and the places you must lift are exactly what this page hunts for. A function f is continuous at a point c when three things hold: f(c) is defined, the limit as x approaches c exists, and the two are equal. Failures come in three flavors students must name: removable discontinuities (a single hole, like (x²−1)/(x−1) at x = 1), jump discontinuities (piecewise definitions that leap, like a step function), and infinite discontinuities (vertical asymptotes, like 1/x at x = 0).

AP Calculus students spend real exam points classifying these, because differentiability requires continuity first — a function cannot have a derivative where it is not continuous. The practical check is one-sided limits: if the left-hand and right-hand limits disagree, you have a jump; if they agree but the point is missing, the hole is removable. Enter the function values around the point and the calculator compares the pieces to deliver the verdict.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Continuity Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.