If a continuously differentiable function
v : I [right arrow] X satisfies the differential inequality
where x = [[[x.sub.1], [x.sub.2], [x.sub.3]].sup.T] x [f.sub.1] ([x.sub.1], [x.sub.2]) and [f.sub.3] (x, [[delta].sup.*.sub.e]) are completely unknown continuously differentiable functions
where [[PHI].sub.i]([p.sub.i], y) and f([p.sub.i], y) are continuously differentiable functions
Because of this reason, if the original function is C1-smooth and monotonic but its derivative is a continuous nowhere differentiable function
, then our rational quadratic FIF r is an ideal tool to approximate such function instead of the classical rational quadratic interpolant, whose derivative is a piece-wise smooth function.
For any differentiable function
y = f(x), if x = x(l) is not only a [omega]-dimensional volume but also describes the length of a [omega]-dimensional fractal curve [gamma].sub.[omega]](l), then the fractal derivative of y = f(x) with respect to the fractal curve [gamma].sub.[omega]] (l) is defined as
where the differentiable functions
f, [g.sub.i]: [R.sup.n] [right arrow] R, i [member of] [m.bar], are invex with respect to the same function [eta]: [R.sup.n] x [R.sup.n] [right arrow] [R.sup.n], the Karush-Kuhn-Tucker necessary optimality conditions are also sufficient.
where [F.sub.1] and [F.sub.1] are differentiable functions
on M, then M is said to be a generalized complex space form (see  and ).
If a network has differential activation functions, then the activations of the output units become differentiable functions
of input variables, the weights and bias.
(Levine [1,2]) Let [PSI](t) be a positive, twice differentiable function
which satisfies for t> 0, the inequality
where a(Y) is an unknown, continuous, positive, and differentiable function
. Singh demonstrates that under his conditions:
Let 8 be an infinitely differentiable function
with compact support satisfying [theta]([xi]) = 1 for [xi] [member of] [D.sub.1].
for any differentiable function
h satisfying certain regularity conditions (see Stein, 1973, 1981).