excluded), then the difference between the values for these arguments will be less than
. This is to hold for any
, however small; in that case the function has a limit for approaches from below. Similarly we define the case when there is a limit for approaches from above. These two limits, even when both exist, need not be identical; and if they are identical, they still need not be identical with the value for the argument
. It is only in this last case that we call the function continuous for the argument
.
A function is called "continuous" (without qualification) when it is continuous for every argument.