places, or everywhere. Examples will be found in any book on the theory of functions of a real variable.
Proceeding now to seek a precise definition of what is meant by saying that a function is continuous for a given argument, when argument and value are both real numbers, let us first define a "neighbourhood" of a number
as all the numbers from
to
, where
is some number which, in important cases, will be very small. It is clear that continuity at a given point has to do with what happens in any neighbourhood of that point, however small.