In mathematics, a property of functions and their graphs. A continuous function is one whose graph has no breaks, gaps, or jumps. It is defined using the concept of a limit. Specifically, a function is said to be continuous at a value x if the limit of the function exists there and is equal to the function's value at that point. When this condition holds true for all real number values of x in an interval, the result is a graph that can be drawn over that interval without lifting the pencil. Such functions are crucial to the theory of calculus, not just because they model most physical systems but because the theorems that lead to the derivative and the integral assume the continuity of the functions involved.