Question

Formal proof question here, mates. Question attached.

Answer 1

???

Answer 2

http://amc.maa.org/a-activities/a7-problems/putnam/-pdf/2010.pdf Problem B5.
My sketch of a solution is:
Let h(y) be the inverse of f(x) for all y >= y0. Then:\[x = h(f(x))\]Differentiang both sides, we get:\[1 = h \prime (f(x))f(f(x))\]That implies that\[h \prime (y) = \frac{1}{f(y)}\]Since h must take on arbitrarily large values, then:\[\int\limits_{y_{0}}^{\infty} \frac{dy}{f(y)}\]must diverge for some boundary of f(y).

Answer 3

My question is, should I set a tight boundary or any boundary. Say,\[f(x) \ge \gamma x^{2}\]is enough but it's not the tighest boundary.

Answer 4

There's a minor error in what I wrote. I meant that that integral must diverge, but it does not for large enough f(y). So I would guess that any boundary works, since it will already contradict the assumption that such function exists.