A function on the set of real numbers is considered to be continuous if its graph on the Cartesian plane is a smooth curve with no “holes” or “jumps.” In other words, any arbitrarily small change to the function’s input corresponds to an arbitrarily small change in its output.

A function is said to be *continuous at a point*, if there is no hole or jump at that point. If there is, the point is called a *discontinuity*. So, for a function to be continuous, it must be continuous at every point in its domain.

This definition is usually sufficient to judge a function’s continuity, but it is not mathematically rigorous. More formally, we can say that if *f* is function mapping *I*, a subset of the real numbers, to *R*, the set of real numbers:

Then *f *is continuous at a point *c*, if the limit of *f(x)* as *x* approaches *c* exists and is equal to *f(c)*. In mathematical notation:

There are 3 rules implicit in this definition:

*f*is defined at*c,*- the limit as
*x*approaches*c*from the lefthand side of the domain must exist, - and that limit must equal
*f(c)*.