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So I have a function $f(x,y,z)$ where $x,y\in \mathbb{R}$ and $z \in \mathbb{R} \cup \{\infty \}$. Now, I want to write something like $f:\mathbb{R}^3 \rightarrow \mathbb{R}$. But the thing is that $z$ includes $\infty$...So what is the correct way of mapping the function $f$ given the domains of $x,y,z$?

Thanks!

Johnny Que
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2 Answers2

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I don't think there is a short notation, but this will do:

$f: \mathbb{R} \times \mathbb{R} \times (\mathbb{R}\cup \{\infty\}) \to \mathbb{R}$

Note that I didn't write the domain as $\mathbb{R²} \times (\mathbb{R}\cup\{\infty\})$, as this would mean the elements in the domain have the form $((a,b),c)$ instead of $(a,b,c)$

However, because there is a $1-1$ correspondence between the two sets, some would argue that $\mathbb{R²} \times (\mathbb{R}\cup\{\infty\})$ is correct too.

  • Thanks! And what if instead of $x,y$, we have a vector $x \in \mathbb{R}^n$...what would be the correct representation? Would it be correct to say $f:\mathbb{R}^n \times (\mathbb{R} \cup {\infty })$? – Johnny Que Dec 17 '17 at 12:35
  • Yes, this is correct. The input has the form $((x_1, \dots, x_n),y)$ –  Dec 17 '17 at 15:29
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You could define $\overline{\mathbb{R}}=\mathbb{R}\cup\{\pm\infty\}$ and $\overline{\mathbb{R}}_{+}=\mathbb{R}\cup\{\infty\}$, then you could write the domain as $\mathbb{R}\times\mathbb{R}\times\overline{\mathbb{R}}_{+}$ and the codomain as $\overline{\mathbb{R}}$ to make sure that functions like $f(x,y,z)=z$ or $f(x,y,z)=-z$ are well-defined.

The Phenotype
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