Is there an elementary continuous function which is positive only if all arguments are?

Question

I am looking for a continuous function $f:\mathbb R^n \to \mathbb R$ such that $f(x_1, x_2, \ldots , x_n) > 0$ if and only if $x_i > 0 \ \forall i = 1,2, \ldots , n$. Can anyone suggest a good one?

score 5 · Accepted Answer · answered Sep 28 '13 at 10:15

5

$f(x)=\displaystyle \min_i x_i$ is simple and continuous.

It is essentially the same as pseudoDust's answer, with $\lt$ changed to $\le$.

answered Sep 28 '13 at 10:15

Henry

157,058

score 2 · Answer 2 · answered Sep 28 '13 at 10:04

2

It's not a nice one, but:

$f(x)=\left\{\begin{matrix} x_0 \; \; \; \; \; \; x_0<x_1,x_2,...,x_n\\ x_1 \; \; \; \; \; \; x_1<x_0,x_2,...,x_n\\ ...\\ x_n \; \; \; \; \; \; x_n<x_0,x_1,...,x_{n-1} \end{matrix}\right.$

answered Sep 28 '13 at 10:04

pseudoDust

228

This is ill defined. What is $f(a,\ldots ,a)$? – Git Gud Sep 28 '13 at 10:07
@GitGud I'll let you guess... – pseudoDust Sep 28 '13 at 11:25
It was a rethorical question. – Git Gud Sep 28 '13 at 11:26

Is there an elementary continuous function which is positive only if all arguments are?

2 Answers2