continuously differentiable

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1: 18.32 OP’s with Respect to Freud Weights

… ►where $Q(x)$ is real, even, nonnegative, and continuously differentiable. …

2: 4.12 Generalized Logarithms and Exponentials

… ►Both $\varphi (x)$ and $\psi (x)$ are continuously differentiable. …

3: 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 $n\ge 1$, $f$ is continuously differentiable on $I$. … … ►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …

4: 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 …

5: 1.5 Calculus of Two or More Variables

… ►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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Symbols:

$\partial x$: partial differential of $x$

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http://dlmf.nist.gov/1.5.E6

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Annotations for §1.5(i), §1.5 and Ch.1

… ►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 $[c,d]$. …

6: 1.8 Fourier Series

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

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: 3.8 Nonlinear Equations

… ►This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: …