We are now ready to define the concept of a function being
continuous. The idea is that we want to say that a function is
continuous if you can draw its graph without taking your pencil
off the page. But sometimes this will be true for some parts of
a graph but not for others. Therefore, we want to start by
defining what it means for a function to be continuous at one
point. The definition is simple, now that we have the concept of
limits: Note that for to be continuous at , the definition in
effect requires three conditions: - that is defined at , so
exists, - the limit as approaches exists, and - the limit and
are equal. If any of these do not hold then is not continuous at
. The idea of the definition is that the point of the graph
corresponding to will be close to the points of the graph
corresponding to nearby -values. Now we can define what it means
for a function to be continuous in general, not just at one
point. We often use the phrase "the function is continuous" to
mean that the function is continu