EDB — 1QH

[1QH] Prerequisites:[1CB]↺↻.

Let \(I\subseteq {\mathbb {R}}\) be an open interval.

Let \(F:I× ℝ→ (0,∞)\) be a positive continuous function, and let \(f: I→ℝ\) be a differentiable function that solves the differential equation

Prove that \(x\) is, either always increasing, in which case \(f'(x)=\sqrt{F(x,f(x))}\) for every \(x\), or it is always decreasing, in which case \(f'(x)=-\sqrt{F(x,f(x))}\); therefore \(f\) is of class \(C^ 1\).