The limit of a continuous function at an endpoint to determine continuity at endpoints.

At this point we have a small problem. We’ve discussed continuity for points in open intervals; in particular, for values where you can look at points “nearby” the point of interest on both the right and left sides. Consider functions such as , the natural domain is . This is not an open interval. What does it mean to say that is continuous at when is not defined for ? To get us out of this quagmire, we need a new definition:

This allows us to talk about continuity on closed and half-closed intervals.

Intuitively this means that a function is called continuous if it is continuous at all the points in its domain; with the understanding that we mean left or right continuous for “end-points” of the domain (if the end-point is included in the domain).

Here we give the graph of a function defined on . Select all intervals for which the following statement is true.

The function is continuous on the interval .

Notice that if we didn’t have the definitions for left and right continuity, then we would always have to include the endpoints of a domain as points of discontinuity; which should seem intuitively wrong. Thus these definitions are really a way to fill in the holes of our definition that our intuition have detected. This is a common process in mathematics; to build definitions in a way to fully encompass what we are trying to describe; and sometimes that requires small sub-definitions - really an extension of the core idea - to fill some special cases.