A continuous function is a function whose graph is a single unbroken curve without any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. A function is continuous on an open interval if the interval is contained in the domain of the function, and the function is continuous at every point of the interval. A function that is continuous on the interval (the whole real line) is often called simply a continuous function; one says also that such a function is continuous everywhere. In mathematical terms, we can define a continuous function using limits. A function f(x) is said to be a continuous function in calculus at a point x = a if the curve of the function does not break at the point x = a. The mathematical definition of the continuity of a function is as follows: A function f(x) is continuous at a point x = a if f(a) exists, limₓ → ₐ f(x) = f(a), and limₓ → ₐ₋ f(x) = limₓ → ₐ₊ f(x) .