Nonexistence of a real analytic function

Question

My question is:

Suppose $f$ is a real analytic function on $\mathbb{R}^d$ and $\|f\|_\infty=1$. $$S=\cup_{k=1}^d l_k,$$ here $$l_k=\Big\{x\in\mathbb{R}^d| x=(0,\cdots,x^k,0\cdots,0),\ x_k\in\mathbb{R} \ \mbox{is the}\ k\mbox{-coordinate}\ \Big\}.$$ If we know, $$f|_S\equiv 1.$$ Can we get $f\equiv1$ on $\mathbb{R}^d$?

Thank you a lot if you can give me any comment or reference.

Answer 1

How about $f(x) = 1 - \prod_k \sin^2(x_k)$?