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).