1 Algebraic and Analytic MethodsTopics of Discussion1.3 Determinants, Linear Operators, and Spectral Expansions1.5 Calculus of Two or More Variables

- §1.4(i) Monotonicity
- §1.4(ii) Continuity
- §1.4(iii) Derivatives
- §1.4(iv) Indefinite Integrals
- §1.4(v) Definite Integrals
- §1.4(vi) Taylor’s Theorem for Real Variables
- §1.4(vii) Maxima and Minima
- §1.4(viii) Convex Functions

If $f({x}_{1})\le f({x}_{2})$ for every pair ${x}_{1}$, ${x}_{2}$ in an interval $I$ such that
$$, then $f(x)$ is *nondecreasing* on $I$. If the $\le $ sign is
replaced by $$, 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.

A function $f(x)$ is *continuous on the right* (or *from above*)
at $x=c$ if

1.4.1 | $$f(c+)\equiv \underset{x\to c+}{lim}f(x)=f(c),$$ | ||

that is, for every arbitrarily small positive constant $\u03f5$ there exists $\delta $ ($>0$) such that

1.4.2 | $$ | ||

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

1.4.3 | $$f(c-)\equiv \underset{x\to c-}{lim}f(x)=f(c).$$ | ||

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, $f(x)=(\mathrm{sin}x)/x$ with $c=0$.

A *simple discontinuity*
of $f(x)$ at $x=c$ occurs when $f(c+)$ and $f(c-)$ exist, but $f(c+)\ne f(c-)$. 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

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

1.4.4 | $${f}^{\prime}(x)=\frac{df}{dx}=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}.$$ | ||

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

1.4.5 | $${(f+g)}^{\prime}(x)={f}^{\prime}(x)+{g}^{\prime}(x),$$ | ||

1.4.6 | $${(fg)}^{\prime}(x)={f}^{\prime}(x)g(x)+f(x){g}^{\prime}(x),$$ | ||

1.4.7 | $${\left(\frac{f}{g}\right)}^{\prime}(x)=\frac{{f}^{\prime}(x)g(x)-f(x){g}^{\prime}(x)}{{(g(x))}^{2}}.$$ | ||

1.4.8 | $${f}^{(2)}(x)=\frac{{d}^{2}f}{{dx}^{2}}=\frac{d}{dx}\left(\frac{df}{dx}\right),$$ | ||

1.4.9 | $${f}^{(n)}={f}^{(n)}(x)=\frac{d}{dx}{f}^{(n-1)}(x).$$ | ||

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

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

1.4.10 | $${h}^{\prime}(x)={f}^{\prime}(g(x)){g}^{\prime}(x).$$ | ||

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

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 | $$f(b)-f(a)=(b-a){f}^{\prime}(c).$$ | ||

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

1.4.12 | $${(fg)}^{(n)}={f}^{(n)}g+\left(\genfrac{}{}{0.0pt}{}{n}{1}\right){f}^{(n-1)}{g}^{\prime}+\mathrm{\cdots}+\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){f}^{(n-k)}{g}^{(k)}+\mathrm{\cdots}+f{g}^{(n)}.$$ | ||

1.4.13 | $$\frac{{d}^{n}}{{dx}^{n}}f(g(x))=\sum \left(\frac{n!}{{m}_{1}!{m}_{2}!\mathrm{\cdots}{m}_{n}!}\right){f}^{(k)}(g(x)){\left(\frac{{g}^{\prime}(x)}{1!}\right)}^{{m}_{1}}{\left(\frac{{g}^{\prime \prime}(x)}{2!}\right)}^{{m}_{2}}\mathrm{\dots}{\left(\frac{{g}^{(n)}(x)}{n!}\right)}^{{m}_{n}},$$ | ||

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

If

1.4.14 | $$\underset{x\to a}{lim}f(x)=\underset{x\to a}{lim}g(x)=0\text{(or}\mathrm{\infty}\text{)},$$ | ||

then

1.4.15 | $$\underset{x\to a}{lim}\frac{f(x)}{g(x)}=\underset{x\to a}{lim}\frac{{f}^{\prime}(x)}{{g}^{\prime}(x)}$$ | ||

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

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

1.4.16 | $$\int fgdx=\left(\int fdx\right)g-\int \left(\int fdx\right)\frac{dg}{dx}dx.$$ | ||

1.4.17 | $$\int {x}^{n}dx=\{\begin{array}{cc}\frac{{x}^{n+1}}{n+1}+C,\hfill & n\ne -1,\hfill \\ \mathrm{ln}\left|x\right|+C,\hfill & n=-1.\hfill \end{array}$$ | ||

For the function $\mathrm{ln}$ see §4.2(i).

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

1.4.18 | $${\int}_{a}^{b}f(x)dx=lim\sum _{j=0}^{n-1}f({\xi}_{j})({x}_{j+1}-{x}_{j})$$ | ||

as $\mathrm{max}({x}_{j+1}-{x}_{j})\to 0$. 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 | $${\int}_{a}^{b}(cf(x)+dg(x))dx=c{\int}_{a}^{b}f(x)dx+d{\int}_{a}^{b}g(x)dx,$$ | ||

$c$ and $d$ constants.

1.4.20 | $${\int}_{a}^{b}f(x)dx=-{\int}_{b}^{a}f(x)dx.$$ | ||

1.4.21 | $${\int}_{a}^{b}f(x)dx={\int}_{a}^{c}f(x)dx+{\int}_{c}^{b}f(x)dx.$$ | ||

1.4.22 | $${\int}_{a}^{\mathrm{\infty}}f(x)dx=\underset{b\to \mathrm{\infty}}{lim}{\int}_{a}^{b}f(x)dx.$$ | ||

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

1.4.23 | $${\int}_{a}^{b}f(x)dx=\underset{c\to b-}{lim}{\int}_{a}^{c}f(x)dx.$$ | ||

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

A generalization of the Riemann integral is the
*Stieltjes integral* ${\int}_{a}^{b}f(x)d\alpha (x)$, where $\alpha (x)$
is a nondecreasing function on the closure of $(a,b)$, which may be bounded, or unbounded, and $d\alpha (x)$ 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 ${C}^{\mathrm{\infty}}$, 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* $d\mu (x)$ and it is
well defined for functions $f$ which are integrable with respect to that more general
measure. See McDonald and Weiss (1999).

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

1.4.23_1 | $$\alpha (d)-\alpha (c)={\int}_{c}^{d}w(x)dx,$$ | ||

$[c,d]\subset I$. | |||

Then

1.4.23_2 | $${\int}_{a}^{b}f(x)d\alpha (x)={\int}_{a}^{b}f(x)w(x)dx,$$ | ||

$f$ integrable with respect to $d\alpha $. | |||

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).

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 $d\alpha (x)$ is
$\delta \left(x-{x}_{n}\right)dx$, where $\delta \left(x-{x}_{n}\right)$ is the Dirac $\delta $-function of §1.17, such that, for $f(x)$ a continuous
function on $(a,b)$, ${\int}_{a}^{b}f(x)d\alpha (x)=f({x}_{n})$ 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 $d\alpha (x)$ could represent a sum, an integral, or a combination of the two. Let $d\alpha (x)=w(x)dx+{\sum}_{n=1}^{N}{w}_{n}\delta \left(x-{x}_{n}\right)dx$, ${x}_{n}\in (a,b)$, $n=1,\mathrm{\dots}N$. Then for $f(x)$ continuous on $(a,b)$,

1.4.23_3 | $${\int}_{a}^{b}f(x)d\alpha (x)={\int}_{a}^{b}w(x)f(x)dx+\sum _{n=1}^{N}{w}_{n}f({x}_{n}).$$ | ||

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.

Let $c\in (a,b)$ and assume that ${\int}_{a}^{c-\u03f5}f(x)dx$ and ${\int}_{c+\u03f5}^{b}f(x)dx$ exist when $$, but not necessarily when $\u03f5=0$. Then we define

1.4.24 | $${\u2a0d}_{a}^{b}f(x)dx=\mathit{P}{\int}_{a}^{b}f(x)dx=\underset{\u03f5\to 0+}{lim}\left({\int}_{a}^{c-\u03f5}f(x)dx+{\int}_{c+\u03f5}^{b}f(x)dx\right),$$ | ||

when this limit exists.

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

1.4.25 | $${\u2a0d}_{-\mathrm{\infty}}^{\mathrm{\infty}}f(x)dx=\mathit{P}{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}f(x)dx=\underset{b\to \mathrm{\infty}}{lim}{\int}_{-b}^{b}f(x)dx,$$ | ||

when this limit exists.

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

1.4.26 | $${\int}_{a}^{b}f(x)dx=F(b)-F(a),$$ | ||

1.4.27 | $$\frac{d}{dx}{\int}_{a}^{x}f(t)dt=f(x).$$ | ||

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

1.4.28 | $${\int}_{a}^{b}f(\varphi (x)){\varphi}^{\prime}(x)dx={\int}_{\varphi (a)}^{\varphi (b)}f(t)dt.$$ | ||

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

1.4.29 | $${\int}_{a}^{b}f(x)\varphi (x)dx=f(c){\int}_{a}^{b}\varphi (x)dx.$$ | ||

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

1.4.30 | $${\int}_{a}^{b}f(x)\varphi (x)dx=f(a){\int}_{a}^{c}\varphi (x)dx+f(b){\int}_{c}^{b}\varphi (x)dx.$$ | ||

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

1.4.31 | $${\int}_{a}^{b}d{x}_{n}{\int}_{a}^{{x}_{n}}d{x}_{n-1}\mathrm{\cdots}{\int}_{a}^{{x}_{2}}d{x}_{1}{\int}_{a}^{{x}_{1}}f(x)dx=\frac{1}{n!}{\int}_{a}^{b}{(b-x)}^{n}f(x)dx.$$ | ||

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

1.4.32 | $$ | ||

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

1.4.33 | $${\mathcal{V}}_{a,b}\left(f\right)=sup\sum _{j=1}^{n}\left|f({x}_{j})-f({x}_{j-1})\right|,$$ | ||

where the supremum is over all sets of points $$ in the
*closure* of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are
finite. If $$, 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 | $${\mathcal{V}}_{a,b}\left(f\right)={\int}_{a}^{b}\left|{f}^{\prime}(x)\right|dx,$$ | ||

whenever this integral exists.

Lastly, whether or not the real numbers $a$ and $b$ satisfy $$, 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).

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

1.4.35 | $$f(x)=\sum _{k=0}^{n}\frac{{f}^{(k)}(a)}{k!}{(x-a)}^{k}+{R}_{n},$$ | ||

1.4.36 | $${R}_{n}=\frac{{f}^{(n+1)}(c)}{(n+1)!}{(x-a)}^{n+1},$$ | ||

$$, | |||

and

1.4.37 | $${R}_{n}=\frac{1}{n!}{\int}_{a}^{x}{(x-t)}^{n}{f}^{(n+1)}(t)dt.$$ | ||

If $f(x)$ is twice-differentiable, and if also ${f}^{\prime}({x}_{0})=0$ and $$ ($>0$), 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 $b$.

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

1.4.38 | $$f((1-t)c+td)\le (1-t)f(c)+tf(d)$$ | ||

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