Let $S$ be a regular surface. And let $f:S\to \mathbb R$ be a differentiable function 
It's not hard to prove that if $ W$ is an open set of $\mathbb R^3$ such $ V\subset S\subset W$, and $f:W\mathbb \to R$ is a differentiable function (in the clssical way) then $ f\left| {_V } \right. $ is differentiable.
I want to prove the converse, i.e all the differentiable functions on $S$ are restrictions of differentiable functions, more precisely:
Let $f: S\to \mathbb R$ be a differentiable function (in this sense of surfaces) then, $f$ is the restriction of a differentiable function $g:W\to\mathbb R$ (differential in the classical way).
I want to prove that $f$ can be extended locally to a differentiable function, and with that we are done. That can be used because we have that locally we have an expression of the form $f(x_1,x_2,x_3)= f(x_1(u,v),x_2(u,v),x_3(u,v))=f(x(u,v))$ But I don't know how.
