$f(x)$ and $g(y)$ are functions of $x$ and $y$, respectively, and $f(x) = g(y)$ for all real values of $x$ and $y$. Which one of the following options is *necessarily* TRUE for all $x$ and $y$?

We are given that $f(x) = g(y)$ for all $x, y \in \mathbb{R}$.

• Since the equation holds for all real values of $x$ and $y$, choose an arbitrary fixed value of $y$, say $y = y_0$. Then:

  $$f(x) = g(y_0) = c$$

  where $c$ is a constant. Since $x$ is arbitrary, $f(x)$ must be a constant function equal to $c$ for all $x$.

• Similarly, choosing a fixed $x = x_0$ gives:

  $$g(y) = f(x_0) = c$$

  so $g(y)$ must also be the same constant for all $y$.

Thus, $f(x) = g(y) = \text{constant}$, which makes (B) necessarily true.

• Options (A) and (D) are only true when the constant is zero, which is not necessarily the case for all possible constant values.