piecewise differentiable

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1: 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. …

2: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►For $f(x)$ piecewise continuously differentiable on $[0,\mathrm{\infty })$ …

3: 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 $x\in I_1$. … ►If $f$ is $n+1$ times continuously differentiable, then … ►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 …

4: 1.9 Calculus of a Complex Variable

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Differentiation

►A function $f(z)$ is complex differentiable at a point $z$ if the following limit exists: … ►A function $f(z)$ is said to be analytic (holomorphic) at $z=z_0$ if it is complex differentiable in a neighborhood of $z_0$. … ►An arc $C$ is given by $z(t)=x(t)+\mathrm{i}y(t)$, $a\le t\le b$, where $x$ and $y$ are continuously differentiable. If $x(t)$ and $y(t)$ are continuous and $x^{\prime }(t)$ and $y^{\prime }(t)$ are piecewise continuous, then $z(t)$ defines a contour. …

5: 10.43 Integrals

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On the interval , $x^{-1}g(x)$ is continuously differentiable and each of $xg(x)$ and $xd(x^{-1}g(x))/dx$ is absolutely integrable.

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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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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 … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. … ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime \prime }(x)\right|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. …

7: 1.4 Calculus of One Variable

… ►For an example, see Figure 1.4.1 … ►When this limit exists $f$ is differentiable at $x$. … ► … ►

Mean Value Theorem

… ►Continuity, or piecewise continuity, of $f(x)$ on $[a,b]$ is sufficient for the limit to exist. …

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 $(a,b)$. … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. …

9: 2.3 Integrals of a Real Variable

… ►converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. … ►If, in addition, $q(t)$ is infinitely differentiable on $[0,\mathrm{\infty })$ and … ►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. … ►In addition to (2.3.7) assume that $f(t)$ and $q(t)$ are piecewise continuous (§1.4(ii)) on $(0,\mathrm{\infty })$, and … ►

On $(a,b)$, $p(t)$ and $q(t)$ are infinitely differentiable and $p^{\prime }(t)>0$.

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