What functions satisfy the condition $f(x,y)=g(x)$?

Question

Are there any functions $f(x,y)$ and $g(x)$ that satisfy

(1) $f(x,y)=g(x)$ for all $x \in \mathbb{R}$ and $y\in \mathbb{R}$

(2) $f(x,y)$ is not constant in $y$ for each $x$ (i.e. for each $x$ there is $y_1$ and $y_2$ such that $f(x,y_1) \neq f(x,y_2)$)

My gut feeling is that there are no such functions that satisfy these conditions because for each fixed $x$ the only function that would satisfy (1) is a constant function, which is ruled out by (2).

Answer 1

Suppose there were such a pair $f(x,y)$ and $g(x)$ such that $g(x) = f(x,y)$ for all $x, y$ and which also satisfy (2). Then $g(x) = f(x,y_1)$ and $g(x) = f(x,y_2)$, but $f(x, y_1) \neq f(x,y_2)$, a contradiction.

Answer 2

Suppose there was a $y_1$ and $ y_2$ such that for all $x, f(x,y_1) \ne f(x,y_2) $ then by the first constraint we also have $f(x,y_1) = g(x) = f(x,y_2) $, which is a contradiction.