# §1.4 Calculus of One Variable

## §1.4(i) Monotonicity

If $f(x_{1})\leq f(x_{2})$ for every pair $x_{1}$, $x_{2}$ in an interval $I$ such that $x_{1}, then $f(x)$ is nondecreasing on $I$. If the $\leq$ sign is replaced by $<$, then $f(x)$ is increasing (also called strictly increasing) on $I$. 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 $f(x)$ is continuous on the right (or from above) at $x=c$ if

 1.4.1 $f(c+)\equiv\lim_{x\to c+}f(x)=f(c),$ ⓘ Symbols: $\equiv$: equals by definition Referenced by: §1.4(ii) Permalink: http://dlmf.nist.gov/1.4.E1 Encodings: TeX, pMML, png See also: Annotations for §1.4(ii), §1.4 and Ch.1

that is, for every arbitrarily small positive constant $\epsilon$ there exists $\delta$ ($>0$) such that

 1.4.2 $\left|f(c+\alpha)-f(c)\right|<\epsilon,$ ⓘ Symbols: $\left|\NVar{x}\right|$: absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.4.E2 Encodings: TeX, pMML, png See also: Annotations for §1.4(ii), §1.4 and Ch.1

for all $\alpha$ such that $0\leq\alpha<\delta$. Similarly, it is continuous on the left (or from below) at $x=c$ if

 1.4.3 $f(c-)\equiv\lim_{x\to c-}f(x)=f(c).$ ⓘ Symbols: $\equiv$: equals by definition Referenced by: §1.4(ii) Permalink: http://dlmf.nist.gov/1.4.E3 Encodings: TeX, pMML, png See also: Annotations for §1.4(ii), §1.4 and Ch.1

And $f(x)$ is continuous at $c$ when both (1.4.1) and (1.4.3) apply.

If $f(x)$ is continuous at each point $c\in(a,b)$, then $f(x)$ is continuous on the interval $(a,b)$ and we write $f\in C(a,b)$. If also $f(x)$ is continuous on the right at $x=a$, and continuous on the left at $x=b$, then $f(x)$ is continuous on the interval $[a,b]$, and we write $f(x)\in C[a,b]$.

A removable singularity of $f(x)$ at $x=c$ occurs when $f(c+)=f(c-)$ but $f(c)$ is undefined. For example, $f(x)=(\sin x)/x$ with $c=0$.

A simple discontinuity of $f(x)$ at $x=c$ occurs when $f(c+)$ and $f(c-)$ exist, but $f(c+)\not=f(c-)$. If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. For an example, see Figure 1.4.1

## §1.4(iii) Derivatives

The derivative $f^{\prime}(x)$ of $f(x)$ is defined by

 1.4.4 $f^{\prime}(x)=\frac{\mathrm{d}f}{\mathrm{d}x}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{% h}.$ ⓘ Defines: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ Referenced by: §1.8(ii) Permalink: http://dlmf.nist.gov/1.4.E4 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4 and Ch.1

When this limit exists $f$ is differentiable at $x$.

 1.4.5 $(f+g)^{\prime}(x)=f^{\prime}(x)+g^{\prime}(x),$ ⓘ A&S Ref: 3.3.2 Permalink: http://dlmf.nist.gov/1.4.E5 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4 and Ch.1
 1.4.6 $(fg)^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x),$ ⓘ A&S Ref: 3.3.3 Permalink: http://dlmf.nist.gov/1.4.E6 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4 and Ch.1
 1.4.7 $\left(\frac{f}{g}\right)^{\prime}(x)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)% }{(g(x))^{2}}.$ ⓘ A&S Ref: 3.3.4 Permalink: http://dlmf.nist.gov/1.4.E7 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4 and Ch.1

### Higher Derivatives

 1.4.8 $f^{(2)}(x)=\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}x}^{2}}=\frac{\mathrm{d}}{% \mathrm{d}x}\left(\frac{\mathrm{d}f}{\mathrm{d}x}\right),$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ Permalink: http://dlmf.nist.gov/1.4.E8 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1
 1.4.9 $f^{(n)}=f^{(n)}(x)=\frac{\mathrm{d}}{\mathrm{d}x}f^{(n-1)}(x).$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$ and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.4.E9 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^{n}(I)$. When $n\geq 1$, $f$ is continuously differentiable on $I$. When $n$ is unbounded, $f$ is infinitely differentiable on $I$ and we write $f\in C^{\infty}(I)$.

### Chain Rule

For $h(x)=f(g(x))$,

 1.4.10 $h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x).$ ⓘ A&S Ref: 3.3.5 Permalink: http://dlmf.nist.gov/1.4.E10 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

### Maxima and Minima

A necessary condition that a differentiable function $f(x)$ has a local maximum (minimum) at $x=c$, that is, $f(x)\leq f(c)$, ($f(x)\geq f(c)$) in a neighborhood $c-\delta\leq x\leq c+\delta$ ($\delta>0$) of $c$, is $f^{\prime}(c)=0$.

### Mean Value Theorem

If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists a point $c\in(a,b)$ such that

 1.4.11 $f(b)-f(a)=(b-a)f^{\prime}(c).$ ⓘ Permalink: http://dlmf.nist.gov/1.4.E11 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

If $f^{\prime}(x)\geq 0$ ($\leq 0$) ($=0$) for all $x\in(a,b)$, then $f$ is nondecreasing (nonincreasing) (constant) on $(a,b)$.

### Leibniz’s Formula

 1.4.12 $(fg)^{(n)}=f^{(n)}g+\genfrac{(}{)}{0.0pt}{}{n}{1}f^{(n-1)}g^{\prime}+\dots+% \genfrac{(}{)}{0.0pt}{}{n}{k}f^{(n-k)}g^{(k)}+\dots+fg^{(n)}.$ ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $k$: integer and $n$: nonnegative integer A&S Ref: 3.3.8 Permalink: http://dlmf.nist.gov/1.4.E12 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

### Faà Di Bruno’s Formula

 1.4.13 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}f(g(x))=\sum\left(\frac{n!}{m_{1}!m_% {2}!\cdots m_{n}!}\right)f^{(k)}(g(x))\*\left(\frac{g^{\prime}(x)}{1!}\right)^% {m_{1}}\left(\frac{g^{\prime\prime}(x)}{2!}\right)^{m_{2}}\dots\left(\frac{g^{% (n)}(x)}{n!}\right)^{m_{n}},$

where the sum is over all nonnegative integers $m_{1},m_{2},\dots,m_{n}$ that satisfy $m_{1}+2m_{2}+\dots+nm_{n}=n$, and $k=m_{1}+m_{2}+\dots+m_{n}$.

### L’Hôpital’s Rule

If

 1.4.14 $\lim\limits_{x\to a}f(x)=\lim\limits_{x\to a}g(x)=0\;\;\mbox{(or \infty)},$ ⓘ Permalink: http://dlmf.nist.gov/1.4.E14 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

then

 1.4.15 $\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f^{\prime}(x)}{g^{\prime}(x)}$ ⓘ A&S Ref: 3.4.1 Referenced by: §1.4(iii), Erratum (V1.1.11) for Subsection 1.4(iii) Permalink: http://dlmf.nist.gov/1.4.E15 Encodings: TeX, pMML, png See also: Annotations for §1.4(iii), §1.4(iii), §1.4 and Ch.1

when the last limit exists. We do assume that $g^{\prime}(x)\neq 0$ for all $x$ in some neighborhood of $a$ with $x\neq a$.

## §1.4(iv) Indefinite Integrals

If $F^{\prime}(x)=f(x)$, then $\int f\,\mathrm{d}x=F(x)+C$, where $C$ is a constant.

### Integration by Parts

 1.4.16 $\int fg\,\mathrm{d}x=\left(\int f\,\mathrm{d}x\right)g-\int\left(\int f\,% \mathrm{d}x\right)\frac{\mathrm{d}g}{\mathrm{d}x}\,\mathrm{d}x.$
 1.4.17 $\int x^{n}\,\mathrm{d}x=\begin{cases}\dfrac{x^{n+1}}{n+1}+C,&\quad n\not=-1,\\ \ln\left|x\right|+C,&\quad n=-1.\end{cases}$

For the function $\ln$ see §4.2(i).

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 (2015), Gröbner and Hofreiter (1949, 1950), and Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

## §1.4(v) Definite Integrals

### Riemann Integrals

Suppose $f(x)$ is defined on $[a,b]$. Let $a=x_{0}, and $\xi_{j}$ denote any point in $[x_{j},x_{j+1}]$, $j=0,1,\dots,n-1$. Then

 1.4.18 $\int^{b}_{a}f(x)\,\mathrm{d}x=\lim\sum^{n-1}_{j=0}f(\xi_{j})(x_{j+1}-x_{j})$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $j$: integer, $n$: nonnegative integer and $\xi_{\NVar{j}}$: point Referenced by: §1.4(v) Permalink: http://dlmf.nist.gov/1.4.E18 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

as $\max(x_{j+1}-x_{j})\to 0$. If the limit exists then $f$ is called Riemann integrable. Continuity, or piecewise continuity, of $f(x)$ on $[a,b]$ is sufficient for the limit to exist.

 1.4.19 $\int^{b}_{a}(cf(x)+dg(x))\,\mathrm{d}x=c\int^{b}_{a}f(x)\,\mathrm{d}x+d\int^{b% }_{a}g(x)\,\mathrm{d}x,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E19 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

$c$ and $d$ constants.

 1.4.20 $\int^{b}_{a}f(x)\,\mathrm{d}x=-\int^{a}_{b}f(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E20 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1
 1.4.21 $\int^{b}_{a}f(x)\,\mathrm{d}x=\int^{c}_{a}f(x)\,\mathrm{d}x+\int^{b}_{c}f(x)\,% \mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Permalink: http://dlmf.nist.gov/1.4.E21 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Infinite Integrals

 1.4.22 $\int^{\infty}_{a}f(x)\,\mathrm{d}x=\lim_{b\to\infty}\int^{b}_{a}f(x)\,\mathrm{% d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §1.4(v), §1.5(v) Permalink: http://dlmf.nist.gov/1.4.E22 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Similarly for $\int^{a}_{-\infty}$. Next, if $f(b)=\pm\infty$, then

 1.4.23 $\int^{b}_{a}f(x)\,\mathrm{d}x=\lim_{c\to b-}\int^{c}_{a}f(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §1.4(v), §1.5(v) Permalink: http://dlmf.nist.gov/1.4.E23 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Similarly when $f(a)=\pm\infty$.

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

### Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals

A generalization of the Riemann integral is the Stieltjes integral $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)$, where $\alpha(x)$ is a nondecreasing function on the closure of $(a,b)$, which may be bounded, or unbounded, and $\,\mathrm{d}\alpha(x)$ is the Stieltjes measure. See Riesz and Sz.-Nagy (1990, Ch. 3). Stieltjes integrability for $f$ with respect to $\alpha$ can be defined similarly as Riemann integrability in the case that $\alpha(x)$ is differentiable with respect to $x$; a generalization follows below.

For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are analytic functions of their variables, and thus $C^{\infty}$, and well defined for all values of these variables; possible exceptions being at boundary points.

A more general concept of integrability of a function on a bounded or unbounded interval is Lebesgue integrability, which allows discussion of functions which may not be well defined everywhere (especially on sets of measure zero) for $x\in\mathbb{R}$. see Rudin (1966), and often used in more abstract mathematical discussions. Similarly the Stieltjes integral can be generalized to a Lebesgue–Stieltjes integral with respect to the Lebesgue-Stieltjes measure $\,\mathrm{d}\mu(x)$ and it is well defined for functions $f$ which are integrable with respect to that more general measure. See McDonald and Weiss (1999).

### Absolutely Continuous Stieltjes Measure

For $\alpha(x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $\,\mathrm{d}\alpha$ is absolutely continuous if $\alpha(x)$ is continuous and there exists a weight function $w(x)\geq 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that

 1.4.23_1 $\alpha(d)-\alpha(c)=\int_{c}^{d}w(x)\,\mathrm{d}x,$ $[c,d]\subset I$. ⓘ Symbols: $[\NVar{a},\NVar{b}]$: closed interval, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\subset$: is contained in and $w(x)$: weight function Referenced by: §1.4(v), Erratum (V1.2.0) §1.4 Permalink: http://dlmf.nist.gov/1.4.E23_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

Then

 1.4.23_2 $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=\int_{a}^{b}f(x)w(x)\,\mathrm{d}x,$ $f$ integrable with respect to $\,\mathrm{d}\alpha$. ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $w(x)$: weight function Referenced by: §1.4(v) Permalink: http://dlmf.nist.gov/1.4.E23_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

In particular, absolute continuity occurs if the function $\alpha(x)$ is differentiable, $\alpha^{\prime}(x)=w(x)$ with $w(x)$ continuous.

For historical reasons, $w(x)$ is also sometimes referred to as a density, as, for example, the mass per unit length at point $x$, see Shohat and Tamarkin (1970, p vii).

### Stieltjes Measure with $\alpha(x)$ Discontinuous

The utility of the generalization implicit in the Stieltjes measure appears when $\alpha(x)$ is not everywhere continuous, but has discontinuous jumps at specific values of $x$, say $x_{n}\in(a,b)$. See Riesz and Sz.-Nagy (1990, Ch. 3). If, for example, $\alpha(x)=H\left(x-x_{n}\right)$, the Heaviside unit step-function (1.16.14), then the corresponding measure $\,\mathrm{d}\alpha(x)$ is $\delta\left(x-x_{n}\right)\,\mathrm{d}x$, where $\delta\left(x-x_{n}\right)$ is the Dirac $\delta$-function of §1.17, such that, for $f(x)$ a continuous function on $(a,b)$, $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=f(x_{n})$ for $x_{n}\in(a,b)$ and $0$ otherwise. Delta distributions and Dirac $\delta$-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17.

Definite integrals over the Stieltjes measure $\,\mathrm{d}\alpha(x)$ could represent a sum, an integral, or a combination of the two. Let $\,\mathrm{d}\alpha(x)=w(x)\,\mathrm{d}x+\sum_{n=1}^{N}w_{n}\delta\left(x-x_{n}% \right)\,\mathrm{d}x$, $x_{n}\in(a,b)$, $n=1,\dots N$. Then for $f(x)$ continuous on $(a,b)$,

 1.4.23_3 $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=\int_{a}^{b}w(x)f(x)\,\mathrm{d}x+\sum_{% n=1}^{N}w_{n}f(x_{n}).$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $n$: nonnegative integer and $w(x)$: weight function Referenced by: §1.4(v), Erratum (V1.2.0) §1.4 Permalink: http://dlmf.nist.gov/1.4.E23_3 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

In the literature where $w(x)$ is considered to be a mass density, the $x_{n}$ are often referred to as mass points, $w_{n}$ being the mass at that point. Ismail (2005, p 5) refers to these $x_{n}$ as isolated atoms.

### Cauchy Principal Values

Let $c\in(a,b)$ and assume that $\int_{a}^{c-\epsilon}f(x)\,\mathrm{d}x$ and $\int_{c+\epsilon}^{b}f(x)\,\mathrm{d}x$ exist when $0<\epsilon<\min(c-a,b-c)$, but not necessarily when $\epsilon=0$. Then we define

 1.4.24 $\pvint^{b}_{a}f(x)\,\mathrm{d}x=P\int^{b}_{a}f(x)\,\mathrm{d}x=\lim_{\epsilon% \to 0+}\left(\int^{c-\epsilon}_{a}f(x)\,\mathrm{d}x+\int^{b}_{c+\epsilon}f(x)% \,\mathrm{d}x\right),$ ⓘ Defines: $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §1.18(vi) Permalink: http://dlmf.nist.gov/1.4.E24 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

when this limit exists.

Similarly, assume that $\int_{-b}^{b}f(x)\,\mathrm{d}x$ exists for all finite values of $b$ ($>0$), but not necessarily when $b=\infty$. Then we define

 1.4.25 $\pvint^{\infty}_{-\infty}f(x)\,\mathrm{d}x=P\int^{\infty}_{-\infty}f(x)\,% \mathrm{d}x=\lim_{b\to\infty}\int^{b}_{-b}f(x)\,\mathrm{d}x,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value Referenced by: §1.14(i) Permalink: http://dlmf.nist.gov/1.4.E25 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

when this limit exists.

### Fundamental Theorem of Calculus

For $F^{\prime}(x)=f(x)$ with $f(x)$ continuous,

 1.4.26 $\int^{b}_{a}f(x)\,\mathrm{d}x=F(b)-F(a),$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §18.17(i) Permalink: http://dlmf.nist.gov/1.4.E26 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1
 1.4.27 $\frac{\mathrm{d}}{\mathrm{d}x}\int^{x}_{a}f(t)\,\mathrm{d}t=f(x).$

### Change of Variables

If $\phi^{\prime}(x)$ is continuous or piecewise continuous, then

 1.4.28 $\int^{b}_{a}f(\phi(x))\phi^{\prime}(x)\,\mathrm{d}x=\int^{\phi(b)}_{\phi(a)}f(% t)\,\mathrm{d}t.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\phi(\NVar{x})$: function Permalink: http://dlmf.nist.gov/1.4.E28 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### First Mean Value Theorem

For $f(x)$ continuous and $\phi(x)\geq 0$ and integrable on $[a,b]$, there exists $c\in[a,b]$, such that

 1.4.29 $\int^{b}_{a}f(x)\phi(x)\,\mathrm{d}x=f(c)\int^{b}_{a}\phi(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\phi(\NVar{x})$: function Permalink: http://dlmf.nist.gov/1.4.E29 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Second Mean Value Theorem

For $f(x)$ monotonic and $\phi(x)$ integrable on $[a,b]$, there exists $c\in[a,b]$, such that

 1.4.30 $\int^{b}_{a}f(x)\phi(x)\,\mathrm{d}x=f(a)\int^{c}_{a}\phi(x)\,\mathrm{d}x+f(b)% \int^{b}_{c}\phi(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\phi(\NVar{x})$: function Permalink: http://dlmf.nist.gov/1.4.E30 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Repeated Integrals

If $f(x)$ is continuous or piecewise continuous on $[a,b]$, then

 1.4.31 $\int_{a}^{b}\,\mathrm{d}x_{n}\int_{a}^{x_{n}}\,\mathrm{d}x_{n-1}\cdots\int_{a}% ^{x_{2}}\,\mathrm{d}x_{1}\int_{a}^{x_{1}}f(x)\,\mathrm{d}x=\frac{1}{n!}\int_{a% }^{b}(b-x)^{n}f(x)\,\mathrm{d}x.$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral and $n$: nonnegative integer Referenced by: §1.15(vi), §1.4(v) Permalink: http://dlmf.nist.gov/1.4.E31 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

### Square-Integrable Functions

A function $f(x)$ is square-integrable if

 1.4.32 $\|f\|^{2}_{2}\equiv\int^{b}_{a}{\left|f(x)\right|}^{2}\,\mathrm{d}x<\infty.$

### Functions of Bounded Variation

With $a, the total variation of $f(x)$ on a finite or infinite interval $(a,b)$ is

 1.4.33 $\mathcal{V}_{a,b}\left(f\right)=\sup\sum^{n}_{j=1}\left|f(x_{j})-f(x_{j-1})% \right|,$ ⓘ Defines: $\mathcal{V}\left(\NVar{f}\right)$: total variation and $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$: total variation Symbols: $\sup$: least upper bound (supremum), $j$: integer, $n$: nonnegative integer and $\left|\NVar{x}\right|$: absolute value of $\NVar{x}$ Permalink: http://dlmf.nist.gov/1.4.E33 Encodings: TeX, pMML, png See also: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

where the supremum is over all sets of points $x_{0} in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. If $\mathcal{V}_{a,b}\left(f\right)<\infty$, then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)=\mathcal{V}_{a,x}\left(f\right)$ and $h(x)=\mathcal{V}_{a,x}\left(f\right)-f(x)$ are nondecreasing bounded functions and $f(x)=g(x)-h(x)$.

If $f(x)$ is continuous on the closure of $(a,b)$ and $f^{\prime}(x)$ is continuous on $(a,b)$, then

 1.4.34 $\mathcal{V}_{a,b}\left(f\right)=\int^{b}_{a}\left|f^{\prime}(x)\right|\,% \mathrm{d}x,$

whenever this integral exists.

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

## §1.4(vi) Taylor’s Theorem for Real Variables

If $f(x)\in C^{n+1}[a,b]$, then

 1.4.35 $f(x)=\sum^{n}_{k=0}\frac{f^{(k)}(a)}{k!}(x-a)^{k}+R_{n},$ ⓘ Symbols: $!$: factorial (as in $n!$), $k$: integer, $n$: nonnegative integer and $R_{\NVar{n}}$: remainder A&S Ref: 3.6.4 Permalink: http://dlmf.nist.gov/1.4.E35 Encodings: TeX, pMML, png See also: Annotations for §1.4(vi), §1.4 and Ch.1
 1.4.36 $R_{n}=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1},$ $a, ⓘ Symbols: $!$: factorial (as in $n!$), $n$: nonnegative integer and $R_{\NVar{n}}$: remainder A&S Ref: 3.6.5 Permalink: http://dlmf.nist.gov/1.4.E36 Encodings: TeX, pMML, png See also: Annotations for §1.4(vi), §1.4 and Ch.1

and

 1.4.37 $R_{n}=\frac{1}{n!}\int^{x}_{a}(x-t)^{n}f^{(n+1)}(t)\,\mathrm{d}t.$

## §1.4(vii) Maxima and Minima

If $f(x)$ is twice-differentiable, and if also $f^{\prime}(x_{0})=0$ and $f^{\prime\prime}(x_{0})<0$ ($>0$), then $x=x_{0}$ is a local maximum (minimum) (§1.4(iii)) of $f(x)$. The overall maximum (minimum) of $f(x)$ on $[a,b]$ will either be at a local maximum (minimum) or at one of the end points $a$ or $b$.

## §1.4(viii) Convex Functions

A function $f(x)$ is convex on $(a,b)$ if

 1.4.38 $f((1-t)c+td)\leq(1-t)f(c)+tf(d)$ ⓘ Permalink: http://dlmf.nist.gov/1.4.E38 Encodings: TeX, pMML, png See also: Annotations for §1.4(viii), §1.4 and Ch.1

for any $c,d\in(a,b)$, and $t\in[0,1]$. See Figure 1.4.2. A similar definition applies to closed intervals $[a,b]$.

If $f(x)$ is twice differentiable, then $f(x)$ is convex iff $f^{\prime\prime}(x)\geq 0$ on $(a,b)$. A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.