# continuous function

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##### 1: 1.4 Calculus of One Variable
###### §1.4(ii) Continuity Figure 1.4.1: Piecewise continuous function on [ a , b ) . Magnify If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^{n}(I)$. …
##### 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 $u\to\infty$
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. …
##### 5: 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 $\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}% \phi(x)\,\mathrm{d}x=\phi(a),$
1.17.9 $\int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(x-a)t}% \,\mathrm{d}t\right)\phi(x)\,\mathrm{d}x=\phi(a).$
##### 6: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
##### 7: 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 $f^{\prime}(x)$. …
##### 8: 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. …
##### 9: 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 $x\in I_{1}$. …
##### 10: 2.1 Definitions and Elementary Properties
For example, suppose $f(x)$ is continuous and $f(x)\sim x^{\nu}$ as $x\to+\infty$ in $\mathbb{R}$, where $\nu$ ($\in\mathbb{C}$) is a constant. …
2.1.11 $\int_{x}^{\infty}f(t)\,\mathrm{d}t\sim-\frac{x^{\nu+1}}{\nu+1},$ $\Re\nu<-1$,
2.1.12 $\int f(x)\,\mathrm{d}x\sim\begin{cases}\text{a constant,}&\Re\nu<-1,\\ \ln x,&\phantom{\Re}\nu=-1,\\ x^{\nu+1}/(\nu+1),&\Re\nu>-1.\end{cases}$