If for every pair , in an interval such that , then is nondecreasing on . If the sign is replaced by , then is increasing (also called strictly increasing) on . Similarly for nonincreasing and decreasing (strictly decreasing) functions. Each of the preceding four cases is classified as monotonic; sometimes strictly monotonic is used for the strictly increasing or strictly decreasing cases.
A function is continuous on the right (or from above) at if
that is, for every arbitrarily small positive constant there exists () such that
for all such that . Similarly, it is continuous on the left (or from below) at if
If is continuous at each point , then is continuous on the interval and we write . If also is continuous on the right at , and continuous on the left at , then is continuous on the interval , and we write .
A removable singularity of at occurs when but is undefined. For example, with .
A simple discontinuity of at occurs when and exist, but . If is continuous on an interval save for a finite number of simple discontinuities, then is piecewise (or sectionally) continuous on . For an example, see Figure 1.4.1
The derivative of is defined by
When this limit exists is differentiable at .
If exists and is continuous on an interval , then we write . When , is continuously differentiable on . When is unbounded, is infinitely differentiable on and we write .
A necessary condition that a differentiable function has a local maximum (minimum) at , that is, , () in a neighborhood () of , is .
If is continuous on and differentiable on , then there exists a point such that
If () () for all , then is nondecreasing (nonincreasing) (constant) on .
where the sum is over all nonnegative integers that satisfy , and .
when the last limit exists. We do assume that for all in some neighborhood of with .
If , then , where is a constant.
For the function see §4.2(i).
Suppose is defined on . Let , and denote any point in , . Then
as . Continuity, or piecewise continuity, of on is sufficient for the limit to exist.
Similarly for . Next, if , then
Similarly when .
Let and assume that and exist when , but not necessarily when . Then we define
when this limit exists.
Similarly, assume that exists for all finite values of (), but not necessarily when . Then we define
when this limit exists.
For with continuous,
If is continuous or piecewise continuous, then
For continuous and and integrable on , there exists , such that
For monotonic and integrable on , there exists , such that
If is continuous or piecewise continuous on , then
A function is square-integrable if
With , the total variation of on a finite or infinite interval is
where the supremum is over all sets of points in the closure of , that is, with added when they are finite. If , then is of bounded variation on . In this case, and are nondecreasing bounded functions and .
If is continuous on the closure of and is continuous on , then
whenever this integral exists.
If , then
If is twice-differentiable, and if also and (), then is a local maximum (minimum) (§1.4(iii)) of . The overall maximum (minimum) of on will either be at a local maximum (minimum) or at one of the end points or .
A function is convex on if
for any , and . See Figure 1.4.2. A similar definition applies to closed intervals .
If is twice differentiable, then is convex iff on . A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.