piecewise continuous

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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 ►

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Figure 1.4.1: Piecewise continuous function on $[a,b)$. Magnify

… ►If ${\varphi }^{\prime }(x)$ is continuous or piecewise continuous, then … ►If $f(x)$ is continuous or piecewise continuous on $[a,b]$, then …

3: 1.8 Fourier Series

… ►If $f(x)$ is of period $2\pi$, and $f^{(m)}(x)$ is piecewise continuous, then … ►For $f(x)$ piecewise continuous on $[a,b]$ and real $\lambda$, … ►If $a_n$ and $b_n$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi ]$, then …

4: 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 $x\in I_1$. … ►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

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

6: 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 $(a,b)$. …

7: 18.18 Sums

… ►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 $(-\mathrm{\infty },\mathrm{\infty })$. …

8: 1.17 Integral and Series Representations of the Dirac Delta

… ►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 $n$. …

9: 2.3 Integrals of a Real Variable

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

10: 10.43 Integrals

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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

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