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Mathematical Expression Editor
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:
A function is left continuous at a point if .
A function is right continuous at a point if .
This allows us to talk about continuity on closed and half-closed intervals.
A function is
continuous on a closed interval if is continuous on , right continuous at , and left continuous at ;
continuous on a half-closed interval if is continuous on and right continuous at ;
continuous on a half-closed interval if is is continuous on and left continuous at .
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.
