I have been trying to develop a function that can combine two probabilities using the rules:

$f(x,y)\in C^\infty (\mathbb{R}^{2})$

$f(x,y)=f(y,x)$

$f(x,1-x)=\frac{1}{2}$

$f(0,x)=0$

$f(x,\frac{1}{2})=x$

$f(x,1)=1$

$f_x(x,y)\geq 0$

I do not believe any polynomial solutions exist. I am wondering if a solution exists and if so how to find it. I belive that if such a solution exists, it would be of the form, or of a similar form, to $a+b^{c}$ where a, b, and c are linear or quadratic functions of x and y, and maybe quartic at worst.