Continuity is a concept in calculus that describes the smoothness of a function. A function is said to be continuous at a point if there is no break in its graph at that point. More formally, a function f(x) is continuous at a point c if and only if the following three conditions are satisfied:

• f(c) is defined
• The limit of f(x) as x approaches c exists
• The limit of f(x) as x approaches c is equal to f(c)

If any of these conditions are not met, the function is said to be discontinuous at that point. Continuity can also be defined on an interval, with slightly different conditions. There are three basic types of discontinuities: removable, jump, and infinite. Removable discontinuities occur when a function has a hole at a point, but can be made continuous by defining the function at that point. Jump discontinuities occur when the left and right limits of a function at a point exist, but are not equal. Infinite discontinuities occur when the left or right limit of a function at a point is infinite.