[adament]

For those who find the existence of such analytic functions intuitively "wrong", note that it is essential to this example that though the second derivative is positive, it is positive and decreasing towards 0. If the second derivative is bounded from below by some positive value, then eventually the first derivative will become positive and it will diverge towards positive infinity and thus the function itself will diverge towards positive infinity.

More precisely suppose that f is twice continuously differentiable and f'(x) < 0 and f''(x) ≥ 0 for all x greater than some K (for example K=0 in the examples given) then λ( (f'')^-1((a, ∞)) ∩ (K, ∞) ) < ∞ for all a > 0 where λ is the Lebesque measure.