# piecewise continuous

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## 1—10 of 15 matching pages

##### 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 If $\phi^{\prime}(x)$ 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: 1.14 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 …
##### 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: 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 $n$. …
##### 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 $(0,\infty)$. … Assume $f(x)$ is real and continuous and $f^{\prime}(x)$ is piecewise continuous on $(-\infty,\infty)$. …
##### 8: 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 $(a,b)$. …
##### 9: 1.17 Integral and Series Representations of the Dirac Delta
More generally, assume $\phi(x)$ is piecewise continuous1.4(ii)) when $x\in[-c,c]$ for any finite positive real value of $c$, and for each $a$, $\int_{-\infty}^{\infty}e^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x$ converges absolutely for all sufficiently large values of $n$. …
##### 10: 2.3 Integrals of a Real Variable
In addition to (2.3.7) assume that $f(t)$ and $q(t)$ are piecewise continuous1.4(ii)) on $(0,\infty)$, and …