# piecewise continuous

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## 1—10 of 15 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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►If $f(x)$ is continuous or piecewise continuous on $[a,b]$, then
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##### 3: 1.8 Fourier Series

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►If $f(x)$ is of period $2\pi $, and ${f}^{(m)}(x)$ is piecewise continuous, then
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►For $f(x)$
piecewise continuous on $[a,b]$ and real $\lambda $,
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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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##### 4: 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 $x\in {I}_{1}$. … ►Sufficient conditions for the limit to exist are that $f(x,y)$ is continuous, or piecewise continuous, on $R$. … ►Moreover, if $a,b,c,d$ are finite or infinite constants and $f(x,y)$ is piecewise continuous on the set $(a,b)\times (c,d)$, then …##### 5: 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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##### 6: 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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##### 7: 18.18 Sums

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►Alternatively, assume $f(x)$ is real and continuous and ${f}^{\prime}(x)$ is piecewise continuous on $(-1,1)$.
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►Assume $f(x)$ is real and continuous and ${f}^{\prime}(x)$ is piecewise continuous on $(0,\mathrm{\infty})$.
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►Assume $f(x)$ is real and continuous and ${f}^{\prime}(x)$ is piecewise continuous on $(-\mathrm{\infty},\mathrm{\infty})$.
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##### 8: 1.17 Integral and Series Representations of the Dirac Delta

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►More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{\mathrm{e}}^{-n{(x-a)}^{2}}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$.
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##### 9: 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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##### 10: 10.43 Integrals

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