It is well known that under suitable conditions, a function which is:

a polynomial in each variable separately is a polynomial in all its variables jointly.

a rational function in each variable separately is a rational function.

a holomorphic function in each variable separately is holomorphic in all its variables.

A complete analytic function can be single-valued or multiple-valued according as it does not have, or does have, branch points. The algebraic functions are examples of the latter.

Here is my question: *is a complete analytic function, which is finitely multiple-valued in each variable separately, also finitely multiple-valued jointly?*

`$(f(x_0,y_0))$`

could take an infinite number of values, if you were to change only one variable in a continuous manner (as in moving around a branch point) and returning to`$(x_0,y_0)$`

, you can reach only a finite number of the values of`$(f(x_0,y_0))$`

, whereas you can reach an infinite number by moving both coordinates together? – Harald Hanche-Olsen Feb 11 '10 at 17:11