What does it mean for a function to be continuous at a point?

A function f is continuous at a point x = a when three conditions all hold. First, f(a) must be defined, meaning the function actually has a value there. Second, the limit of f(x) as x approaches a must exist, which requires the left hand limit and the right hand limit to be equal. Third, that common limit value must equal the function value f(a). In short, the limit exists and matches what the function actually does at the point. Intuitively, the graph has no break, hole, or jump there, so you can draw through the point without lifting your pen. To check in an exam, compute the left limit, the right limit, and f(a) separately, then confirm all three are equal. If any differs, the function is discontinuous at that point.