## The Difference Between Continuity and The Existence of a Limit of a Function at a Point

Briefly, a function is continuous at a point if is close to for sufficiently close to We make this precise in the following definition.

Definition

Suppose and If then is continuous at if and only if for each there is a such that if then If is continuous at for every then we say is continuous.

Continuity and limits are not the same thing. For a function to be continuous at a point the function must be defined at that point. A function need not be defined at a point for to have a limit at that point.

For example, The function is not defined at 0 since but since so the function has a limit at x=0 but is not continuous at 0.

A more extreme, clearer, example is given by Then but This is because only has to tend to 0 without ever being equal to 0. In fact the value of at a point may have no relation to the limit of as This function also has a limit at but is not continuous at since there exists no such that for all since if and  