differentiable function


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differentiable function

[‚dif·ə′ren·chə·bəl ′fəŋk·shən]
(mathematics)
A function which has a derivative at each point of its domain.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Here we consider the case of infinitely differentiable functions on [R.sub.+] with the condition that ([[f.sup.m])[sigma])(x) is uniformly bounded with respect to m and x.
Let V: [[t.sub.0], [infinity]) x [R.sup.n] [right arrow] [R.sup.+] be continuously differentiable function. The solution of the fractional differential equation
where [alpha] is a real parameter which will be determined later and v is a twice continuously differentiable function. That is, u(x, t) depends on x and t primarily through the term [absolute value of x]/ [absolute value of 4kt].
A differentiable function [phi] : S [subset or equal to] [R.sup.n] [right arrow] [R.sup.n] is said to be (strictly) pseudoinvex function with respect to [eta] : S x S [right arrow] [R.sup.n] if
Then, for any [sigma] [member of] G, a neighborhood U [subset] [R.sup.l] of [[??].sub.0] and a neighborhood G [subset] [R.sup.r] of [[sigma].sub.0] can make the equation [PSI]([??], [sigma]) = 0 which has a unique solution [??] [member of] U, and the solution can be expressed as [??] = [g.sub.0] ([sigma]), where [g.sub.0](*) is a continuously differentiable function on [sigma] = [[sigma].sub.0].
Then, for any differentiable function w : [0,n] [intersection] Z [right arrow] R, and each k = 1,2, ..., n, one has
where f(x) and y(x) are differentiable functions. Here, we apply fractional integration operational matrix of rational Haar wavelet to solve Abel integral equations as fractional integral equations.
where h(t) = diag{[h.sub.1](t), [h.sub.2](t), ..., [h.sub.n](t)}, [h.sub.i](t) are continuously differentiable functions, and h(t) is a scaling function matrix.
Let f : I [subset] r [right arrow] R be a twice differentiable function on [I.sup.[omicron]] ([I.sup.[omicron]] is the interior of I), and let a,b [member of] [I.sup.[omicron]] with a < b.
Suppose that f : [a,b] [right arrow] R be a twice differentiable function on (a,b) and suppose that [gamma] [less than or equal to] f" (t) [less than or equal to] [gamma] for all t [member of] (a, b).
Let (Eq.) be a differentiable function on (Eq.) such that (Eq.) where (Eq.).
It is clear that the payoff function [[PHI].sub.i]([p.sub.i], y) is a continuously differentiable function of [p.sub.i](s) and y(s).