Let $\mathbb{D}:= \{(x,y)\in \mathbb{R}^2:x^2+y^2<1\}$ , $S$ is open and $\mathbb{\bar{D}}\subset S$. Let $F:S\to \mathbb{R}$ is differentiable and $|F(x)|\leq 1$ for all $x\in \mathbb{D}$. Prove that there exists $\theta \in S$ such that $|\triangledown F(\theta )| < 4$.
I think it may be easier to solve if it is continuously differentiable. In this condition, $\triangledown F(x)$ can be seen as a $C^1$ vector field and I can find a integral curve originated from $(0,0)$.
When it is differentiable, what I know is only "The differential mean value theorem": $F(y)-F(x)=\triangledown F(\xi )\cdot (y-x)$ . It is hard to consider directions of these two vectors.