R is the reals. f, g : [0, 1] → R are arbitary functions. Show that we can find x, y such that |xy - f(x) - g(y)| ≥ 1/4.
Solution
It is enough to consider the values at 0 and 1. If the pairs (0, 0), (0, 1), (1, 0) do not work for (x, y), then we have |f(0) + g(0)| < 1/4, |f(0) + g(1)| < 1/4, and |f(1) + g(0)| < 1/4. Hence f(1) + g(1) ≤ f(1) + g(0) + f(0) + g(1) - (f(0) + g(0)) < 1/4 + 1/4 + 1/4. So 1 - f(1) - g(1) > 1 - 3/4 = 1/4. Hence the pair (1, 1) does work.
© John Scholes
jscholes@kalva.demon.co.uk
15 Feb 2002