B. Definition of Continuity

Over the last few sections we’ve been using the term “nice enough” to define those functions that we could evaluate limits by just evaluating the function at the point in question. It’s now time to formally define what we mean by “nice enough”.

Definition:

This definition can be turned around into the following fact.

This is exactly the same fact that we first put down back when we started looking at limits with the exception that we have replaced the phrase “nice enough” with continuous.

It’s nice to finally know what we mean by “nice enough”, however, the definition doesn’t really tell us just what it means for a function to be continuous. Let’s take a look at an example to help us understand just what it means for a function to be continuous.

From this example we can get a quick “working” definition of continuity. A function is continuous on an interval if we can draw the graph from start to finish without ever once picking up our pencil. The graph in the last example has only two discontinuities since there are only two places where we would have to pick up our pencil in sketching it.