# piecewise continuous functions

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##### 1: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
##### 2: 1.4 Calculus of One Variable
For an example, see Figure 1.4.1 Figure 1.4.1: Piecewise continuous function on [ a , b ) . Magnify If $\phi^{\prime}(x)$ is continuous or piecewise continuous, then …
##### 3: 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 $(a,b)$. …
##### 4: 1.8 Fourier Series
If $a_{n}$ and $b_{n}$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi]$, then …
##### 5: 1.5 Calculus of Two or More Variables
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
In addition to (2.3.7) assume that $f(t)$ and $q(t)$ are piecewise continuous1.4(ii)) on $(0,\infty)$, and …
##### 7: 1.14 Integral Transforms
If $f(t)$ is continuous and $f^{\prime}(t)$ is piecewise continuous on $[0,\infty)$, then … If $f(t)$ is piecewise continuous, then … Also assume that $f^{(n)}(t)$ is piecewise continuous on $[0,\infty)$. … If $f(t)$ and $g(t)$ are piecewise continuous, then … If $f(t)$ is piecewise continuous on $[0,\infty)$ and the integral (1.14.47) converges, then …
##### 8: 10.43 Integrals
###### §10.43(iii) Fractional Integrals
The Kontorovich–Lebedev transform of a function $g(x)$ is defined as …
• (b)

$g(x)$ is piecewise continuous and of bounded variation on every compact interval in $(0,\infty)$, and each of the following integrals

• ##### 9: 18.18 Sums
###### §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 $(0,\infty)$. … Assume $f(x)$ is real and continuous and $f^{\prime}(x)$ is piecewise continuous on $(-\infty,\infty)$. … For the modified Bessel function $I_{\nu}\left(z\right)$ see §10.25(ii). …
##### 10: 18.2 General Orthogonal Polynomials
A system (or set) of polynomials $\{p_{n}(x)\}$, $n=0,1,2,\ldots$, is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\geq 0$) if
18.2.1 $\int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x=0,$ $n\neq m$.
Here $w(x)$ is continuous or piecewise continuous or integrable, and such that $0<\int_{a}^{b}x^{2n}w(x)\,\mathrm{d}x<\infty$ for all $n$. … More generally than (18.2.1)–(18.2.3), $w(x)\,\mathrm{d}x$ may be replaced in (18.2.1) by a positive measure $\,\mathrm{d}\alpha(x)$, 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 $0<\int_{a}^{b}x^{2n}\,\mathrm{d}\alpha(x)<\infty$ for all $n$. … This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). …