At most points, $S'(x)$ is the length of the right leg minus the length of the left leg of the curvilinear triangle, perhaps with exceptions on a null set where there are tangencies. If $S(0)=S(1)=0$ then the lengths of these legs are at most $\sqrt{2}(1-x)$ and $\sqrt{2}x$. For almost all $0\le x \le 1$, $S'(x)$ satisfies $-\sqrt{2}\le -\sqrt{2} x \lt S'(x) \lt \sqrt(2) (1-x) \le \sqrt{2}$. This is an extra condition on $H$ which rules out some smooth small functions which have large derivatives near some points, such as $10^6 \exp(-1/(x (1-x)))$ 1-x))^2)$for$0\lt x \lt 1$, which has a derivative of$1.132$at$x=0.436$although the value of the function is small. 1 There are restrictions. At most points,$S'(x)$is the length of the right leg minus the length of the left leg of the curvilinear triangle, perhaps with exceptions on a null set where there are tangencies. If$S(0)=S(1)=0$then the lengths of these legs are at most$\sqrt{2}(1-x)$and$\sqrt{2}x$. For almost all$0\le x \le 1$,$S'(x)$satisfies$-\sqrt{2}\le -\sqrt{2} x \lt S'(x) \lt \sqrt(2) (1-x) \le \sqrt{2}$. This is an extra condition on$H$which rules out some smooth small functions which have large derivatives near some points, such as$10^6 \exp(-1/(x (1-x)))$for$0\lt x \lt 1$, which has a derivative of$1.132$at$x=0.436\$ although the value of the function is small.