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1: 6.16 Mathematical Applications

… ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

2: About Color Map

… ►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. …

3: 1.4 Calculus of One Variable

… ►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

… ►The curve $C$ is piecewise differentiable if $𝐜$ 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: 1.5 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 …

6: 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 …

7: 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 …

8: 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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ISSN 0036-1410, Document, MathReview (Kathi Karima Selig), ZentralBlatt

Referenced by:

§16.4(iii).

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9: Bibliography L

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Generalized Riemann $\zeta$-function regularization and Casimir energy for a piecewise uniform string. Phys. Rev. D 44 (2), pp. 560–562.

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Document

Referenced by:

§24.18.

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10: 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 $(a,b)$. …