# differentiable

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## 1—10 of 29 matching pages

##### 1: 18.32 OP’s with Respect to Freud Weights
where $Q(x)$ is real, even, nonnegative, and continuously differentiable. …
##### 2: 1.4 Calculus of One Variable
When this limit exists $f$ is differentiable at $x$. … When $n\geq 1$, $f$ is continuously differentiable on $I$. … …
###### Mean Value Theorem
A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …
##### 3: 1.6 Vectors and Vector-Valued Functions
when $f$ is continuously differentiable. … The curve $C$ is piecewise differentiable if $\mathbf{c}$ is piecewise differentiable. … … For $x$, $y$, and $z$ continuously differentiable, the vectors … For $f$ and $g$ twice-continuously differentiable functions …
##### 4: 4.12 Generalized Logarithms and Exponentials
Both $\phi(x)$ and $\psi(x)$ are continuously differentiable. … For $C^{\infty}$ generalized logarithms, see Walker (1991). …
##### 5: 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 $[0,\infty)$ and … assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. … Assume that $q(t)$ again has the expansion (2.3.7) and this expansion is infinitely differentiable, $q(t)$ is infinitely differentiable on $(0,\infty)$, and each of the integrals $\int e^{ixt}q^{(s)}(t)\,\mathrm{d}t$, $s=0,1,2,\dots$, converges at $t=\infty$, uniformly for all sufficiently large $x$. …
• (a)

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

• ##### 6: 1.13 Differential Equations
Let $W(z)$ satisfy (1.13.14), $\zeta(z)$ be any thrice-differentiable function of $z$, and
1.13.19 $\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}\zeta}^{2}}=\left(\dot{z}^{2}H(z)-\tfrac{1% }{2}\left\{z,\zeta\right\}\right)U.$
1.13.21 $\left\{z,\zeta\right\}=(\ifrac{\mathrm{d}\xi}{\mathrm{d}\zeta})^{2}\left\{z,% \xi\right\}+\left\{\xi,\zeta\right\}.$
1.13.22 $\left\{z,\zeta\right\}=-(\ifrac{\mathrm{d}z}{\mathrm{d}\zeta})^{2}\left\{\zeta% ,z\right\}.$
##### 7: 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 $[c,d]$. …
##### 8: 2.8 Differential Equations with a Parameter
The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to $\xi$. … Again, $u>0$ and $\psi(\xi)$ is $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$. … The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to $\xi$. … The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to $\xi$. … The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to $\xi$. …
##### 9: 18.38 Mathematical Applications
In consequence, expansions of functions that are infinitely differentiable on $[-1,1]$ in series of Chebyshev polynomials usually converge extremely rapidly. …
##### 10: 1.8 Fourier Series
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 $|f^{\prime\prime}(x)|$ are integrable over $(-\infty,\infty)$. …