, the function is continuous for this argument. This may be taken as defining continuity: it is equivalent to our former definition.
We can define the limit of a function for a given argument (if it exists) without passing through the ultimate oscillation and the four limits of the general case. The definition proceeds, in that case, just as the earlier definition of continuity proceeded. Let us define the limit for approaches from below. If there is to be a definite limit for approaches to
from below, it is necessary and sufficient that, given any small number
, two values for arguments sufficiently near to
(but both less than