# continuously differentiable

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## 1—10 of 23 matching pages

##### 1: 4.12 Generalized Logarithms and Exponentials
Both $\phi(x)$ and $\psi(x)$ are continuously differentiable. …
##### 2: 18.32 OP’s with Respect to Freud Weights
where $Q(x)$ is real, even, nonnegative, and continuously differentiable, where $xQ^{\prime}(x)$ increases for $x>0$, and $Q^{\prime}(x)\to\infty$ as $x\to\infty$, see Freud (1969). …
##### 3: 1.5 Calculus of Two or More Variables
The function $f(x,y)$ is continuously differentiable if $f$, $\ifrac{\partial f}{\partial x}$, and $\ifrac{\partial f}{\partial y}$ are continuous, and twice-continuously differentiable if also $\ifrac{{\partial}^{2}f}{{\partial x}^{2}}$, $\ifrac{{\partial}^{2}f}{{\partial y}^{2}}$, $\,{\partial}^{2}f/\,\partial x\,\partial y$, and $\,{\partial}^{2}f/\,\partial y\,\partial x$ are continuous. …
1.5.6 $\frac{\,{\partial}^{2}f}{\,\partial x\,\partial y}=\frac{\,{\partial}^{2}f}{\,% \partial y\,\partial x}.$
If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\ifrac{\partial F}{\partial y}\not=0$ 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 $\ifrac{\partial f}{\partial x}$ are continuous on a rectangle $a\leq x\leq b$, $c\leq y\leq d$; (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
If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^{n}(I)$. When $n\geq 1$, $f$ 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
when $f$ is continuously differentiable. … For $x$, $y$, and $z$ continuously differentiable, the vectors … when $\mathbf{F}$ is a continuously differentiable vector-valued function. … when $\mathbf{F}$ is a continuously differentiable vector-valued function. … For $f$ and $g$ twice-continuously differentiable functions …
##### 6: 18.40 Methods of Computation
In what follows we consider only the simple, illustrative, case that $\mu(x)$ is continuously differentiable so that $\,\mathrm{d}\mu(x)=w(x)\,\mathrm{d}x$, 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). …
##### 7: 9.11 Products
For any continuously-differentiable function $f$
##### 8: 30.14 Wave Equation in Oblate Spheroidal Coordinates
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. …
##### 9: 2.7 Differential Equations
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 …
##### 10: 1.8 Fourier Series
Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime\prime}(x)\right|$ are integrable over $(-\infty,\infty)$. …