Continuity : In the last few sections we've been using the term "nice enough" to describe those functions which we could evaluate limits by just evaluating the function at the point in question.  Now it's time to formally define what we mean by "nice enough".

Definition

A function f ( x ) is called to be continuous at x = a if

A function is called continuous on the interval [a, b] if it is continuous at each of the point in the interval.

Note as well that this definition is also implicitly supposing that both f ( a ) and exist.  If either of these do not present then the function will not be continuous at x = a . This definition can be turned around into the following fact.

Fact 1

If f (x) is continuous at x = a then,

It is exactly the similar fact that first we put down back while we started looking at limits  along with the exception which we have replaced the phrase "nice enough" with continuous.

It's nice to at last know what we mean by "nice enough", however, the definition doesn't actually tell us just what it means for any function to be continuous.  Let's take a look at an instance to help us understand just what it means for a function to be continuous.