On separately polynomial functions (Q805767)

In a recent issue of Am. Math. Mon. [Vol. 97, No.7 (1990)] \textit{J. Totaro} proposed the following problem: Let S be the boundary of the unit square [0,1]\(\times [0,1]\) in \({\mathbb{R}}^ 2\). Suppose f is a continuous real-valued function on S such that f(x,0) and f(x,1) are polynomial functions of x on [0,1] and such that f(0,y) and f(1,y) are polynomial functions of y on [0,1]. Prove that f is the restriction of a polynomial function of x and y. It is the aim of the present note to generalize this problem and then to give a new proof of a result of R. S. Palais on separately polynomial functions.