# defining a function. question about domain and how to specify validity

context:

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.

• @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. – Michael Levy Apr 16 at 23:08

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 $$[0,1]$$.