Exercise 15, § 1

Let $f$ be a real function, defined and continuous for $|t| \leq a, |x| \leq b$, such that $f(t, x) < 0$ for $tx > 0$ and $f(t, x) > 0$ for $tx < 0$; show that $x = 0$ is the only integral of the equation $x' = f(t, x)$ which takes the value 0 at the point $t = 0$ (argue by contradiction).