# piecewise

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## 1—10 of 17 matching pages

##### 1: 6.16 Mathematical Applications

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►It occurs with Fourier-series expansions of all piecewise continuous functions.
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##### 2: About Color Map

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►We therefore use a piecewise linear mapping as illustrated below, that takes phase $0$ to red, $\pi /2$ to yellow, $\pi $ to cyan and $3\pi /2$ to blue.
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##### 3: 1.4 Calculus of One Variable

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►If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is

*piecewise*(or*sectionally*) continuous on $I$. For an example, see Figure 1.4.1 … ►Continuity, or piecewise continuity, of $f(x)$ on $[a,b]$ is sufficient for the limit to exist. … ►If ${\varphi}^{\prime}(x)$ is continuous or piecewise continuous, then … ►If $f(x)$ is continuous or piecewise continuous on $[a,b]$, then …##### 4: 1.6 Vectors and Vector-Valued Functions

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►The curve $C$ is

*piecewise differentiable*if $\mathbf{c}$ is piecewise differentiable. … … ►Sufficient conditions for this result to hold are that ${F}_{1}(x,y)$ and ${F}_{2}(x,y)$ are continuously differentiable on $S$, and $C$ is piecewise differentiable. … ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. …##### 5: 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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##### 6: 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 …##### 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: 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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##### 9: 18.2 General Orthogonal Polynomials

##### 10: Bibliography D

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Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets.
SIAM J. Math. Anal. 30 (5), pp. 1029–1056.
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