Multilinear Functions and Polynomials

[Dragonfall]

A function $$f : \mathbf{R}^n\rightarrow\mathbf{R}$$ is multilinear if it's linear in every variable. Is there a multilinear function that's not a multilinear polynomial?

Given a function defined on the n dimensional hypercube, values of which are 0 or 1, there is a unique multilinear extension to all of R. Is this extension a polynomial?

 

[HallsofIvy]

No, a "linear function" is a polynomial, by definition, so any multilinear function is a polynomial. Saying "multilinear polynomial" is redundant.