Continuity

A function is continuous if you can draw its graph without lifting your pencil — no jumps, no holes, no vertical walls. It's the mathematical version of 'no surprises': the value at a point matches the values nearby, and small changes in x produce small changes in y. Continuity is the gentle condition that makes limits, derivatives, and integrals behave. When it fails, the failure comes in exactly three flavors: a removable hole, a jump, or an infinite blow-up. Naming the break tells you what the function can and can't do there.