i](r + [tau])]} provides the transmit power of n SUs under the condition of grand coalition N, and the continuously

differentiable function W([p.

If f : [a,b] [right arrow] R is a

differentiable function with bounded derivative and [[alpha].

p] can be expressed as a

differentiable function of the output variable [y.

Let 8 be an infinitely

differentiable function with compact support satisfying [theta]([xi]) = 1 for [xi] [member of] [D.

2] be a continuously

differentiable function, and let A be a subset of h ([M.

for any

differentiable function h satisfying certain regularity conditions (see Stein, 1973, 1981).

Furthermore, we also note that for an arbitrary scalar twice

differentiable function f, the following equality is permitted [22]:

In the case of a

differentiable function F(y, z), we use subscript letters to denote derivatives [F.

Let f(x) be a real-valued

differentiable function defined for x [element of] [R.

For mathematical convenience, f(x) must be a continuous and

differentiable function.

5) Using these conventions, a continuous function can only be represented by an uninterrupted or gapless line, a line drawn without lifting the nib off the paper, so to speak; and a

differentiable function can only be represented by a line which is not only uninterrupted but also smooth, without sharp bends or sudden changes of direction.

The basis of the proof was Sard's theorem on the set of critical values of a

differentiable function, of which I learned in the late sixties in a first encounter with Steve Smale, whose path had remained separate from mine during the period of campus turbulence that began in September 1964.

Following the notation of Riley [1975] and Spence [1974], each applicant has an underlying ability level, n, which falls within the interval [n(0), n(1)] and the proportion of the applicant pool with ability less than n is assumed to be a

differentiable function F(n).

D] be the closed unit disc in C and let h be a continuously

differentiable function on [bar.

Let f: [a, b] [subset] R [right arrow] R be a

differentiable function such that f' [member of] L[a, b], Then, for n [member of] N, k > 0, and x [member of] [a, b], one has