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I'm working on a proof. I'm also working to improve my mathematical writing.

questions

Q.1: Is there a more compact, yet formal way of writing the following?

``Let $f\colon \mathbb {R} \to \mathbb {R}$ be the function defined by the equation $f(x) = \sqrt{1-x^2}$, valid for all real values of $x$ greater than or equal to zero and less than or equal to one.''

Q.2:

As written above, what is the domain? Is it $\mathbb {R}$ or the real values $x$ that are greater than or equal to zero and less than or equal to one?

Q.3

Is there anyway to tighten up the following, while preserving or augmenting the formalism?

``For a positive integer $m$, let $f\colon \mathbb{R}^m \to \mathbb {R}$ be the function defined by the equation $f(x_1, \ldots, x_m) = \sqrt{1-\sum\limits_{k=1}^m x_k^2}$, which is valid for all real values of $0\leq x_k \leq 1$.''

Q.4

Rigorously speaking, does the quote in Q.3 mean that $0\leq x_1 \leq 1,\ldots, 0\leq x_M \leq 1$? If not, how can I include this needed language?

Note: I have purposely excluding $x$ that are greater than or equal to -1 and less than or equal to 0.

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  • $\begingroup$ @Peter Foreman, correct me if I am wrong, but this appears not to include x = 0 or x=1. In addition, it is not explicit that x can take all real values between 0 and 1. $\endgroup$ – Michael Levy Apr 16 at 23:08
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Q1: Let $f:\mathbb R \to \mathbb R$ be the function defined by $f(x)=\sqrt {1-x^{2}}$ for $0 \leq x \leq 1$.

Q2: the domain is $[0,1]$.

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