




A sequence is said to approach x_{0} or converge to x0 from below if it converges to x_{0}and all its member are less than x_{0}.
We say that the limit of f from below at x_{0} is h, written as , if for any sequence {xj} that approaches x_{0} from below, {f(xj)} approaches h.
Similarly the limit of f from above at x_{0} is h, written as , if for any sequence {xj} that approaches x_{0} from above, {f(xj)} approaches h.
We write when the limits from above and below are both h.
With these definitions, we can prove that if
then
Thus the limit of sum is the sum of the limits of the terms summed; and the limit of a product is the product of the limits of its factors, (when they exist).
A function f is continuous at x = x_{0} if exists and is f(x_{0}).
A continuous function has the property that it has no "gaps" that is:
if it takes on values a and b, it takes on every value between them.
Most functions we encounter are continuous almost everywhere.