partial differentiation

[rillucas]

I am stuck on problem out the classic Rosenlicht Intro to Analysis book. The problem is:

Let f be a differentiable real-valued function on the open ball in E^n of center (a_1, ...., a_n) and radius r and suppose that f'_n = 0. Prove that there is a unique real-valued function g on the open ball in E^(n-1) of center (a_1,...., a_(n-1)) and radius r such that f(x_1, ..., x_n) = g(x_1, ...., x_(n-1)) and this g is differentiable.

I am attempting to use a Lemma stating that f is differentiable iff f(x)-f(a)=f'_1(x)(x_1-a_1)+...+f'_n(x)(x_n-a_n) and then show that this can be represented by a unique function in E^(n-1) and then show partials exist and are continuous.....but I'm getting stuck even starting the problem. Any help would be greatly appreciated.

Thanks

 

[DrMike]

I'd suggest letting g be the restriction of f to the open ball, perhaps that will get you started, or perhaps that was already obvious/