# §1.4 Calculus of One Variable

## §1.4(i) Monotonicity

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.

## §1.4(ii) Continuity

A function is continuous on the right (or from above) at if

1.4.1

that is, for every arbitrarily small positive constant there exists () such that

1.4.2

for all such that . Similarly, it is continuous on the left (or from below) at if

1.4.3

And is continuous at when both (1.4.1) and (1.4.3) apply.

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

Figure 1.4.1: Piecewise continuous function on .

## §1.4(iii) Derivatives

The derivative of is defined by

1.4.4

When this limit exists is differentiable at .

1.4.5
1.4.6
1.4.7

### ¶ Higher Derivatives

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 .

For ,

1.4.10

### ¶ Maxima and Minima

A necessary condition that a differentiable function has a local maximum (minimum) at , that is, , () in a neighborhood () of , is .

### ¶ Mean Value Theorem

If is continuous on and differentiable on , then there exists a point such that

1.4.11

If () () for all , then is nondecreasing (nonincreasing) (constant) on .

1.4.12

### ¶ Faà Di Bruno’s Formula

where the sum is over all nonnegative integers that satisfy , and .

### ¶ L’Hôpital’s Rule

If

1.4.14

then

1.4.15

when the last limit exists.

## §1.4(iv) Indefinite Integrals

If , then , where is a constant.

### ¶ Integration by Parts

See §§4.10, 4.26(ii), 4.26(iv), 4.40(ii), and 4.40(iv) for indefinite integrals involving the elementary functions.

For extensive tables of integrals, see Apelblat (1983), Bierens de Haan (1867), Gradshteyn and Ryzhik (2000), Gröbner and Hofreiter (1949, 1950), and Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

## §1.4(v) Definite Integrals

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.

1.4.19

and constants.

1.4.20
1.4.21

### ¶ Infinite Integrals

1.4.22

Similarly for . Next, if , then

1.4.23

Similarly when .

When the limits in (1.4.22) and (1.4.23) exist, the integrals are said to be convergent. If the limits exist with replaced by , then the integrals are absolutely convergent. Absolute convergence also implies convergence.

### ¶ Cauchy Principal Values

Let and assume that and exist when , but not necessarily when . Then we define

1.4.24

when this limit exists.

Similarly, assume that exists for all finite values of (), but not necessarily when . Then we define

when this limit exists.

### ¶ Fundamental Theorem of Calculus

For with continuous,

1.4.26

### ¶ Change of Variables

If is continuous or piecewise continuous, then

1.4.28

### ¶ First Mean Value Theorem

For continuous and and integrable on , there exists , such that

1.4.29

### ¶ Second Mean Value Theorem

For monotonic and integrable on , there exists , such that

1.4.30

### ¶ Repeated Integrals

If is continuous or piecewise continuous on , then

### ¶ Square-Integrable Functions

A function is square-integrable if

1.4.32

### ¶ Functions of Bounded Variation

With , the total variation of on a finite or infinite interval is

1.4.33

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

1.4.34

whenever this integral exists.

Lastly, whether or not the real numbers and satisfy , and whether or not they are finite, we define by (1.4.34) whenever this integral exists. This definition also applies when is a complex function of the real variable . For further information on total variation see Olver (1997b, pp. 27–29).

If , then

1.4.35
1.4.36,

and

## §1.4(vii) Maxima and Minima

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 .

## §1.4(viii) Convex Functions

A function is convex on if

1.4.38

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.

Figure 1.4.2: Convex function . , , , .