# continuously differentiable

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## 1—10 of 23 matching pages

##### 1: 4.12 Generalized Logarithms and Exponentials

##### 2: 18.32 OP’s with Respect to Freud Weights

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►where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $x{Q}^{\prime}(x)$ increases for $x>0$, and ${Q}^{\prime}(x)\to \mathrm{\infty}$ as $x\to \mathrm{\infty}$, see Freud (1969).
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##### 3: 1.5 Calculus of Two or More Variables

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►The function $f(x,y)$ is

*continuously differentiable*if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous,*and twice-continuously differentiable*if also ${\partial}^{2}f/{\partial x}^{2}$, ${\partial}^{2}f/{\partial y}^{2}$, ${\partial}^{2}f/\partial x\partial y$, and ${\partial}^{2}f/\partial y\partial x$ are continuous. … ►
1.5.6
$$\frac{{\partial}^{2}f}{\partial x\partial y}=\frac{{\partial}^{2}f}{\partial y\partial x}.$$

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►If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\partial F/\partial y\ne 0$ at $(a,b)$, then in a *neighborhood*of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and ${g}^{\prime}(x)=-{F}_{x}/{F}_{y}$. … ►If $f$ is $n+1$ times continuously differentiable, then … ►Sufficient conditions for validity are: (a) $f$ and $\partial f/\partial x$ are continuous on a rectangle $a\le x\le b$, $c\le y\le d$; (b) when $x\in [a,b]$ both $\alpha (x)$ and $\beta (x)$ are continuously differentiable and lie in $[c,d]$. …##### 4: 1.4 Calculus of One Variable

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►If ${f}^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in {C}^{n}(I)$.
When $n\ge 1$, $f$ is

*continuously differentiable*on $I$. … … ►In particular, absolute continuity occurs if the function $\alpha (x)$ is differentiable, ${\alpha}^{\prime}(x)=w(x)$ with $w(x)$ continuous. … ►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …##### 5: 1.6 Vectors and Vector-Valued Functions

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►when $f$ is continuously differentiable.
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►For $x$, $y$, and $z$
continuously differentiable, the vectors
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►when $\mathbf{F}$ is a continuously differentiable vector-valued function.
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►when $\mathbf{F}$ is a continuously differentiable vector-valued function.
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►For $f$ and $g$ twice-continuously differentiable functions
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##### 6: 18.40 Methods of Computation

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►In what follows we consider only the simple, illustrative, case that $\mu (x)$ is continuously differentiable so that $d\mu (x)=w(x)dx$, with $w(x)$ real, positive, and continuous on a real interval $[a,b].$ The strategy will be to: 1) use the moments to determine the recursion coefficients ${\alpha}_{n},{\beta}_{n}$ of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas ${x}_{i}$ and weights (or Christoffel numbers) ${w}_{i}$ from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32).
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##### 7: 9.11 Products

##### 8: 30.14 Wave Equation in Oblate Spheroidal Coordinates

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►If ${b}_{1}={b}_{2}=0$, then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire $(x,y,z)$-space.
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##### 9: 2.7 Differential Equations

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►In a finite or infinite interval $({a}_{1},{a}_{2})$ let $f(x)$ be real, positive, and twice-continuously differentiable, and $g(x)$ be continuous.
…has twice-continuously differentiable solutions
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##### 10: 1.8 Fourier Series

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►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|{f}^{\prime \prime}(x)\right|$ are integrable over $(-\mathrm{\infty},\mathrm{\infty})$.
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