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I guess they are defining domain in a very general way. So you can say a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is taking values (as a function) in $\mathbb{R}^{\mathbb{R}}$.

Is this a common thing? I read a lot of mathematics (applied generally) and I have not seen this before, at least that I can remember.

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mathtick
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    No, $f"\Bbb R\to\Bbb R$ does not "take values" in $\Bbb R^{\Bbb R}$. Rather, $f\in\Bbb R^{\Bbb R}$. – Angina Seng Jun 24 '20 at 17:11
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    For sets $X$ and $Y$, $Y^X$ is used to denote the set of functions from $X$ to $Y$. – R. Burton Jun 24 '20 at 17:13
  • @AnginaSeng yes, that is what I mean by "take values" in. I guess it is a bit confusing to say "take" for a function. – mathtick Jun 24 '20 at 21:39
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    If $f:A\to B$ then $f$ takes (on) values in $B$, takes (in) inputs from $A$ and is an element of $B^A$ or ${}^AB$. – Mark S. Jun 24 '20 at 23:20

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What I have seen is this: A function $f$ on $\mathbb{R}$ is a relation such that $f :=$ {$(a,b): a \in \mathbb{R}, b \in \mathbb{R}$} $\subseteq \mathbb{R} \times \mathbb{R}$. However, as $f$ is a subset of $\mathbb{R}^2$, we interpret the function $f$ as assigning to every $a$ in $\mathbb{R}$ exactly one $b$ in $\mathbb{R}$ and write $f : \mathbb{R} \to \mathbb{R}$ such that $f(a):=b$. As mentioned in the comments, for any sets $X$ and $Y$, the set of all functions $f : X \to Y$ is denoted $Y^X$.

Taylor Rendon
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