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I'm interested in understanding how many regions can be created in $\mathbb{R}^n$ using $k$ functions $F_1, \ldots, F_k: \mathbb{R}^n \to \mathbb{R}$ that are of class $C^1$ (i.e., continuously differentiable). Specifically, I want to know the maximum number of regions these functions can partition $\mathbb{R}^n$ into.

In the context of hyperplanes given by affine functions, it's known from this post that the maximum number of regions is given by: $$ \sum_{i=0}^n \binom{k}{i} $$

However, my question is: does this formula still apply when the functions $F_i$ are not affine? Additionally, I believe there is a hypothesis that every two hyperplanes should intersect exactly once. How does this hypothesis affect the result when the functions are non-affine?

hanava331
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    There is no upper bound anymore: if you consider the functions $x\mapsto\cos(x)$ and $x\mapsto \sin(x)$, their graphs enclose a (countable) infinite number of regions in the plane. – Balloon Sep 03 '24 at 15:28
  • @Balloon thanks for your response. I agree, but where do $$ \sum_{i=0}^n \binom{k}{i} $$ comes from? do we assume that two hyperplanes do intersect exactly once? – hanava331 Sep 04 '24 at 18:27

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