A question from my Analysis list that I could not have any idea. Any help would be great. I don't want a complete solution, just a little hint, because I need to do at least one part alone.
Let $U = \lbrace x \in \mathbb{R}^{m}\,|\, |x_{i}| < 1, i =1,...,m\rbrace$ and $f: U \longrightarrow \mathbb{R}$ a function differentiable, with $\displaystyle \Bigg| \frac{\partial f}{\partial x_{i}}(x)\Bigg| \leq 3$ for all $x \in U$. Then $f(U)$ is an interval of length $\leq 3m$.
Small hints
For $m=2$ you can write for example: $$f(x_1,x_2)=\int_0^{x_2} \partial_2 f(x_1,s) ds+f(x_1,0)=\int_0^{x_2} \partial_2 f(x_1,s) ds+\int_0^{x_1} \partial_1 f(s,0) ds+f(0,0)$$ and thus: $$|f(x_1,x_2) -f(0,0)| \leq \int_0^{x_2} |\partial_2 f(x_1,s)|ds+\int_0^{x_1} |\partial_1 f(s,0)| ds $$
And to show that a subset of $\Bbb R$ is an interval you only have to show that it is connected.