# continuous function

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## 1—10 of 84 matching pages

##### 1: 1.4 Calculus of One Variable
###### §1.4(ii) Continuity
If $f(x)\in C^{n+1}[a,b]$, then …
##### 2: 4.12 Generalized Logarithms and Exponentials
For $C^{\infty}$ generalized logarithms, see Walker (1991). …
##### 3: 2.8 Differential Equations with a Parameter
in which $\xi$ ranges over a bounded or unbounded interval or domain $\mathbf{\Delta}$, and $\psi(\xi)$ is $C^{\infty}$ or analytic on $\mathbf{\Delta}$. … Again, $u>0$ and $\psi(\xi)$ is $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$. Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that are $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$, and as $u\to\infty$Also, $\psi(\xi)$ is $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$, and $u>0$. … In the former, corresponding to any positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that are $C^{\infty}$ on $(0,\alpha_{2})$, and as $u\to\infty$
##### 4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
and functions $f(x),g(x)\in C^{2}(a,b)$, assumed real for the moment. … For $f(x)\in C(X)\cap L^{2}\left(X\right)\cap\mathcal{D}(T)$, $f(x)$ has the eigenfunction expansion, following directly from (1.18.17)–(1.18.19), … For $f(x)\in C(X)\cap L^{2}\left(X\right)\cap\mathcal{D}(T)$, $f(x)$ has the eigenfunction expansion, analogous to that of (1.18.33), … More generally, for $f\in C(X)$, $x\in X$, see (1.4.24), … , $f\in C^{2}(X)$) of $Lf=zf$ which is moreover in $L^{2}\left(X\right)$. …
where $h=b-a$, $f\in C^{2}[a,b]$, and $a<\xi. … If in addition $f$ is periodic, $f\in C^{k}(\mathbb{R})$, and the integral is taken over a period, then … Let $h=\frac{1}{2}(b-a)$ and $f\in C^{4}[a,b]$. … If $f\in C^{2m+2}[a,b]$, then the remainder $E_{n}(f)$ in (3.5.2) can be expanded in the form … For $C^{\infty}$ functions Gauss quadrature can be very efficient. …
##### 6: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
##### 7: 1.13 Differential Equations
$u$ and $z$ belong to domains $U$ and $D$ respectively, the coefficients $f(u,z)$ and $g(u,z)$ are continuous functions of both variables, and for each fixed $u$ (fixed $z$) the two functions are analytic in $z$ (in $u$). … As the interval $[a,b]$ is mapped, one-to-one, onto $[0,c]$ by the above definition of $t$, the integrand being positive, the inverse of this same transformation allows $\widehat{q}(t)$ to be calculated from $p,q,\rho$ in (1.13.31), $p,\rho\in C^{2}(a,b)$ and $q\in C(a,b)$. …
##### 8: 1.17 Integral and Series Representations of the Dirac Delta
1.17.2 $\int_{-\infty}^{\infty}\delta\left(x-a\right)\phi(x)\,\mathrm{d}x=\phi(a),$ $a\in\mathbb{R}$,
From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that
1.17.6 $\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x% -a)^{2}}\phi(x)\,\mathrm{d}x=\phi(a),$
1.17.9 $\int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}% ^{\mathrm{i}(x-a)t}\,\mathrm{d}t\right)\phi(x)\,\mathrm{d}x=\phi(a).$
##### 9: 3.7 Ordinary Differential Equations
Let $(a,b)$ be a finite or infinite interval and $q(x)$ be a real-valued continuous (or piecewise continuous) function on the closure of $(a,b)$. … If $q(x)$ is $C^{\infty}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. …
##### 10: 1.8 Fourier Series
Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … If $a_{n}$ and $b_{n}$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi]$, then … If a function $f(x)\in C^{2}[0,2\pi]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime}(x)$. …