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Let $f$ and $g$ be (non-constant) functions from $\mathbb{R}^d$ to $\mathbb{R}$. For a point $x \in \mathbb{R}^d$, let us define the set $S_{f,x} = \{y \in \mathbb{R}^d : f(y) = f(x) \land y \neq x \}$, and the set $S_{g,x} = \{y \in \mathbb{R}^d : g(y) = g(x) \land y \neq x \}$.

Is it possible to define $f$ and $g$ such that $\forall x \in \mathbb{R}^d, S_{f,x} \cap S_{g,x} = \emptyset$ ? If not, then how can one define $f$ and $g$ so that (for any $x$) the number of points in $S_{f,x} \cap S_{g,x}$ is minimized ?

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  • $\begingroup$ Take $f:\mathbb{R}^d\to\mathbb{R}$ to be injective. Then $S_{f,x}=\emptyset$ for all $x$. The $g$ can be any non-constant function in that case. $\endgroup$
    – SphericalTriangle
    Commented Mar 21, 2018 at 19:39
  • $\begingroup$ @SphericalTriangle Is it straightforward to find an injective function $f : \mathbb{R}^d \to \mathbb{R}$ ? Do you have some examples (closed form expression) of such function ? $\endgroup$
    – eLearner
    Commented Mar 21, 2018 at 22:06
  • $\begingroup$ @eLearner - most functions do not have "closed form expressions". There are many injective functions from $\Bbb R^d \to \Bbb R$, but they are not nice. One example that maps $(0,1)^d \to \Bbb R$ is obtained by interweaving the digits of the decimal expansions of $x_1, ..., x_n$ to form the value of $f(x_1, ..., x_n)$. $\endgroup$
    – Paul Sinclair
    Commented Mar 22, 2018 at 3:15
  • $\begingroup$ @PaulSinclair It doesn't really matter to me if the functions aren't nice. Do you mean that e.g. for $d=3$ if the input is $< 0.01234, 0.56789423, 0.137 >$ then it should be mapped to the output $0.0511632773849423$ ? And what if the domain is $\mathbb{R}^d$ instead of $(0,1)^d$ ? $\endgroup$
    – eLearner
    Commented Mar 22, 2018 at 8:26
  • $\begingroup$ Yes. That is one possible map. $(0,1)$ and $\Bbb R$ can be easily put in 1-1 correspondence. For example, $\tan \pi\left(x-\dfrac12\right)$ does so. $\endgroup$
    – Paul Sinclair
    Commented Mar 22, 2018 at 16:05

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