# piecewise continuous functions

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## 1—10 of 14 matching pages

##### 1: 6.16 Mathematical Applications

##### 2: 1.4 Calculus of One Variable

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►For an example, see Figure 1.4.1
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►If ${\varphi}^{\prime}(x)$ is continuous or piecewise continuous, then
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##### 3: 3.7 Ordinary Differential Equations

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►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 $(a,b)$.
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##### 4: 1.8 Fourier Series

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►If ${a}_{n}$ and ${b}_{n}$ are the Fourier coefficients of a piecewise continuous function
$f(x)$ on $[0,2\pi ]$, then
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##### 5: 1.5 Calculus of Two or More Variables

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►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 $x\in {I}_{1}$. …##### 6: 2.3 Integrals of a Real Variable

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►In addition to (2.3.7) assume that $f(t)$ and $q(t)$ are piecewise continuous (§1.4(ii)) on $(0,\mathrm{\infty})$, and
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##### 7: 1.14 Integral Transforms

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►If $f(t)$ is continuous and ${f}^{\prime}(t)$ is piecewise continuous on $[0,\mathrm{\infty})$, then
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►If $f(t)$ is piecewise continuous, then
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►Also assume that ${f}^{(n)}(t)$ is piecewise continuous on $[0,\mathrm{\infty})$.
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►If $f(t)$ and $g(t)$ are piecewise continuous, then
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►If $f(t)$ is piecewise continuous on $[0,\mathrm{\infty})$ and the integral (1.14.47) converges, then
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##### 8: 10.43 Integrals

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(b)
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###### §10.43(i) Indefinite Integrals

… ►###### §10.43(iii) Fractional Integrals

… ► … ►The*Kontorovich–Lebedev transform*of a function $g(x)$ is defined as … ►$g(x)$ is piecewise continuous and of bounded variation on every compact interval in $(0,\mathrm{\infty})$, and each of the following integrals

##### 9: 18.18 Sums

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###### §18.18(i) Series Expansions of Arbitrary Functions

… ►Alternatively, assume $f(x)$ is real and continuous and ${f}^{\prime}(x)$ is piecewise continuous on $(-1,1)$. … ►Assume $f(x)$ is real and continuous and ${f}^{\prime}(x)$ is piecewise continuous on $(0,\mathrm{\infty})$. … ►Assume $f(x)$ is real and continuous and ${f}^{\prime}(x)$ is piecewise continuous on $(-\mathrm{\infty},\mathrm{\infty})$. … ►For the modified Bessel function ${I}_{\nu}\left(z\right)$ see §10.25(ii). …##### 10: 18.2 General Orthogonal Polynomials

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►A system (or set) of polynomials $\{{p}_{n}(x)\}$, $n=0,1,2,\mathrm{\dots}$, is said to be

*orthogonal on*$(a,b)$*with respect to the weight function*$w(x)$ ($\ge 0$)*if*►
18.2.1
$${\int}_{a}^{b}{p}_{n}(x){p}_{m}(x)w(x)dx=0,$$
$n\ne m$.

►Here $w(x)$ is continuous or piecewise continuous or integrable, and such that $$ for all $n$.
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►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by a positive measure $d\alpha (x)$, where $\alpha (x)$ is a bounded nondecreasing function on the closure of $(a,b)$ with an infinite number of points of increase, and such that $$ for all $n$.
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►This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)).
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