continuous function

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11: 1.5 Calculus of Two or More Variables

… ►

§1.5(i) Partial Derivatives

►A function

f(x,y)

is continuous at a point

(a,b)

if … ►A function is continuous on a point set

D

if it is continuous at all points of

D

. A function

f(x,y)

is piecewise continuous on

I_{1}\times I_{2}

, where

I_{1}

and

I_{2}

are intervals, if it is piecewise continuous in

x

for each

y\in I_{2}

and piecewise continuous in

y

for each

x\in I_{1}

. …

12: 2.1 Definitions and Elementary Properties

… ►For example, suppose

f(x)

is continuous and

f(x)\sim x^{\nu}

x\to+\infty

\mathbb{R}

, where

\nu

(

\in\mathbb{C}

) is a constant. … ►

2.1.11

\int_{x}^{\infty}f(t)\,\mathrm{d}t\sim-\frac{x^{\nu+1}}{\nu+1},

\Re\nu<-1

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $f(x)$ : continuous function and $\nu$ : complex constant
Permalink:: http://dlmf.nist.gov/2.1.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §2.1(ii), §2.1 and Ch.2

►

2.1.12

\int f(x)\,\mathrm{d}x\sim\begin{cases}\text{a constant,}&\Re\nu<-1,\\ \ln x,&\phantom{\Re}\nu=-1,\\ x^{\nu+1}/(\nu+1),&\Re\nu>-1.\end{cases}

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $f(x)$ : continuous function and $\nu$ : complex constant
Permalink:: http://dlmf.nist.gov/2.1.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §2.1(ii), §2.1 and Ch.2

…

13: 3.11 Approximation Techniques

… ►Furthermore, if

f\in C^{\infty}[-1,1]

, then the convergence of (3.11.11) is usually very rapid; compare (1.8.7) with

k

arbitrary. … ►Let

f

be continuous on a closed interval

[a,b]

and

w

be a continuous nonvanishing function on

[a,b]

w

is called a weight function. …

14: 1.9 Calculus of a Complex Variable

… ►

Continuity

►A function

f(z)

is continuous at a point

z_{0}

\lim\limits_{z\to z_{0}}f(z)=f(z_{0})

. … ►A function

f(z)

is continuous on a region

R

if for each point

z_{0}

R

and any given number

\epsilon

(

>0

) we can find a neighborhood of

z_{0}

such that

\left|f(z)-f(z_{0})\right|<\epsilon

for all points

z

in the intersection of the neighborhood with

R

. … ►Let

(a,b)

be a finite or infinite interval, and

f_{0}(t),f_{1}(t),\dots

be real or complex continuous functions,

t\in(a,b)

. …

15: 1.6 Vectors and Vector-Valued Functions

… ►The path integral of a continuous function

f(x,y,z)

is …If

h(a)=a^{\prime}

and

h(b)=b^{\prime}

, then the reparametrization is called orientation-preserving, and … ►The integral of a continuous function

f(x,y,z)

over a surface

S

is …

16: 1.10 Functions of a Complex Variable

… ►Assume that for each

t\in[a,b]

f(z,t)

is an analytic function of

z

D

, and also that

f(z,t)

is a continuous function of both variables. … ►For each

t\in[a,b)

f(z,t)

is analytic in

D

;

f(z,t)

is a continuous function of both variables when

z\in D

and

t\in[a,b)

; the integral (1.10.18) converges at

b

, and this convergence is uniform with respect to

z

in every compact subset

S

D

. …

17: 28.31 Equations of Whittaker–Hill and Ince

… ►ambiguities in sign being resolved by requiring

C_{p}^{m}(x,\xi)

and

{S_{p}^{m}}^{\prime}(x,\xi)

to be continuous functions of

x

and positive when

x=0

. …

18: 15.6 Integral Representations

… ►In all cases the integrands are continuous functions of

t

on the integration paths, except possibly at the endpoints. …

19: 31.9 Orthogonality

… ►The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. …

20: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous

2\pi

-periodic function

f(x)

whose second derivative is square-integrable over the interval

[0,2\pi]

can be expanded in a uniformly and absolutely convergent series …