gross@spp2.UUCP (Howard E. Gross) (04/12/85)
I believe that the following problem appeared in net.math: Since for
integer x we have x**2 = x + x + ... +x added to itself x-times, why
is it that the drivative of the left hand side is 2x while the
drivative on the rhs is x? Besides the answer that the rhs only is
defined for integer x, another approach is to define a natural
extension of the rhs for all reals x. One such extension is the
following: Define [x] to be the largest integer less than or equal to
x. Define f(x) = x + ... + x added [x] times. Where is f continuous,
on what intervals is it differentiable and what is its derivative?
Are there any more "natural" extentions of the rhs of the first
equation?
A more general question: Let N denote the positive integers, R the
reals. For p: N --> R, g: R --> N, h: R --> R, when is k(x) =
p(i)*h(x) summed from i=1 to g(x) continuous, differentiable and what
is its derivative? What conditions on p,g and h are required?
C
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