How to prove or disprove that the expression is real analyatic.

Question

Consider the surface defined by $$ z=2-2\cos(x+y+z) .$$ Express $z$ as a function of $(x,y)$ near $(0,0,0).$ Is this expression real analytic?

I have solved the first part using implicit function theorem. But how to say whether the expression is real analytic or not. Because using implicit function theorem we can say that we can write $z=f(x,y)$.

Answer 1

Let $F(x,y,z)=z-2+2\cos(x+y+z)$. The implicit function theorem says: there is a neigborhood $U \subset \mathbb R^2$ of $(0,0)$ and exactly one function $f:U \to \mathbb R$ such that

$F(x,y,f(x,y))=0$ for all $(x,y) \in U$ and $f(0,0)=0$.

Furthermore we have, since $F \in C^{\infty}(\mathbb R^3)$,

$$f \in C^{\infty}(U).$$