# piecewise

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##### 1: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
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 1.4.1Continuity, or piecewise continuity, of $f(x)$ on $[a,b]$ is sufficient for the limit to exist. … If $\phi^{\prime}(x)$ 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 $\mathbf{c}$ 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
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 …
##### 6: 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 …
##### 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: 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 $(0,\infty)$. … Assume $f(x)$ is real and continuous and $f^{\prime}(x)$ is piecewise continuous on $(-\infty,\infty)$. …
##### 9: 18.2 General Orthogonal Polynomials
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 $n$. …
##### 10: Bibliography D
• G. C. Donovan, J. S. Geronimo, and D. P. Hardin (1999) Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (5), pp. 1029–1056.