We are given a function $f(x, y)$ defined piecewise as follows: $f(x, y) = \begin{cases} \frac{x^2 - 4y^2}{x - 2y}, & \text{if } x \neq 2y \\ g(x), & \text{if } x = 2y \end{cases}$ We are told that $f$ is continuous in the whole plane and we need to find a formula for $g(x)$.

1. Problem Description

We are given a function $f(x, y)$ defined piecewise as follows:

$f(x, y) = \begin{cases}

\frac{x^2 - 4y^2}{x - 2y}, & \text{if } x \neq 2y \\

g(x), & \text{if } x = 2y

\end{cases}$

We are told that $f$ is continuous in the whole plane and we need to find a formula for $g(x)$.

2. Solution Steps

For $f(x, y)$ to be continuous, the two pieces of the function must agree when $x = 2y$. Therefore, we need to find the limit of $\frac{x^2 - 4y^2}{x - 2y}$ as $x$ approaches $2y$.

We can factor the numerator:

$x^2 - 4y^2 = (x - 2y)(x + 2y)$

Then, we have:

$\frac{x^2 - 4y^2}{x - 2y} = \frac{(x - 2y)(x + 2y)}{x - 2y}$

For $x \neq 2y$, we can cancel the $(x - 2y)$ terms:

$\frac{(x - 2y)(x + 2y)}{x - 2y} = x + 2y$

Now, we need to find the limit as $x$ approaches $2y$:

$\lim_{(x, y) \to (2y, y)} x + 2y$

Since $x = 2y$, we substitute $2y$ for $x$:

$2y + 2y = 4y$

Therefore, $g(x) = x + 2y$ when $x = 2y$. We substitute $x/2$ for $y$.

$g(x) = x + 2(x/2) = x + x = 2x$

To make $f$ continuous, we need to have $g(x) = 2x$.

3. Final Answer

$g(x) = 2x$