2.2 Formal Definition of Limit and Continuity

2.2.1 Limits

A sequence is said to approach x0 or converge to x0 from below if it converges to x0and all its member are less than x0.

We say that the limit of f from below at x0 is h, written as , if for any sequence {xj} that approaches x0 from below, {f(xj)} approaches h.

Similarly the limit of f from above at x0 is h, written as , if for any sequence {xj} that approaches x0 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

Proof of 1

Proof of 2

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).

2.2.2 Continuity

A function f is continuous at x = x0 if exists and is f(x0).

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.

Proof