Let F be an increasing real function defined for all real X, where 0<=X<=1 such that:

(i) F (X/8) = F(X)/7 and
(ii) F(1-X) = 1 – F(X)

For all whole numbers M and N greater than zero, determine:
F ( 1/ ((8^M)* ( 8^N + 1)) ) in terms of M and N.

(In reply to Rigorous Solution by Gamer)

Perhaps instead of "(ie F(64) = F(8)/7 = F(1)/49)"

you mean "(ie F(1/64) = F(1/8)/7 = F(1)/49)"

It seems you allow confusion by not constraining X as you proceed (F(X) is only defined in 0<=X<=1), so if we rewrite rule i as you suggest "7*F(X)=F(8X)" X would now be constrained to 0<=X<=1/8. Obviously, since M and N are positive whole numbers, F(1/(8^M)), F(1/( 8^N + 1)), and F (1/((8^M)*(8^N + 1))) are all always within this range.

I know you suggest this by limiting U when setting U=8X in your proof, but I am not sure that was clear.

Other than that, nice proof!

Posted by Eric on 2006-01-05 23:34:30