The Meaning of Continuity
Briefly, a functionis continuous at a pointifis close toforsufficiently close to
We make this precise in the following definition.
SupposeandIfthenis continuous atif and only if for eachthere is asuch that ifthen
Ifis continuous atfor every then we sayis continuous.
Continuity and limits are not the same thing. For a function to be continuous at a pointthe function must be defined at that point. A function need not be defined at a point forto have a limit at that point.
For example,The function is not defined at 0 sincebut sinceso the function has a limit at x=0 but is not continuous at 0.
A more extreme, clearer, example is given by
ThenbutThis is becauseonly has to tend to 0 without ever being equal to 0. In fact the value ofat a pointmay have no relation to the limit ofasThis function also has a limit atbut is not continuous atsince there exists nosuch thatfor allsince ifand