Let $B \subseteq \mathbb{R}^n$ be a product of closed bounded intervals in $\mathbb{R}$. Fix $N>0$. Suppose I want to cover $B$ with $N$ open sets, $U_1, \ldots, U_N$, and get a smooth partition of unity $\rho_1, \ldots, \rho_N$ with respect to these sets. I was wondering is it possible to do this in a way that the derivatives of the $\rho_j$'s are bounded independent of the choice of the $U_j$'s?

I was wondering maybe I am asking for too much here and that this is not possible, or maybe it's possible? I have no idea... I would appreciate any comments or suggestions. Thank you.

PS I would like to change the question slightly. I would like to assume that each $U_j$ is not too small in that each $U_j$ contains an open set of the form $(x_1 - \varepsilon, x_1+ \varepsilon) \times \cdots \times (x_n - \varepsilon, x_n+ \varepsilon)$ for some $\varepsilon > 0$.

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