A function is said to be continuous at a point if there is no break, jump, or hole in its graph at that point. More formally, a function f(x)f(x)f(x) is continuous at x=ax=ax=a if three conditions are met:

1. The function f(a)f(a)f(a) is defined.
2. The limit of the function as xxx approaches aaa exists: lim⁡x→af(x)\lim_{x\to a}f(x)limx→a​f(x) exists.
3. The limit value equals the function value at that point: lim⁡x→af(x)=f(a)\lim_{x\to a}f(x)=f(a)limx→a​f(x)=f(a).

If these conditions hold true at every point in an interval, the function is continuous over that interval. This means you could draw its graph without lifting your pencil, illustrating no interruptions or breaks in the curve. Continuity is a fundamental concept in calculus because a function must be continuous at a point to be differentiable there. It ensures the function behaves predictably and smoothly near that point.