To prove that $f(x, y)$ is continuous at $(x_0, y_0)$, we need to show that:

$\lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)$

Given that $f(x, y)$ is differentiable at $(x_0, y_0)$, we know the following definition holds:

$f(x_0 + \Delta x, y_0 + \Delta y) = f(x_0, y0) + \Delta x f_x(x_0, y_0) + \Delta y f_y(x_0, y_0) + \epsilon(\Delta x, \Delta y)$

Where $\epsilon(\Delta x, \Delta y)$ is a function that satisfies:

$\lim_{(\Delta x, \Delta y) \to (0, 0)} \frac{\epsilon(\Delta x, \Delta y)}{\sqrt{(\Delta x)^2 + (\Delta y)^2}} = 0$

This property implies that the function $\epsilon(\Delta x, \Delta y)$ tends to zero faster than $\sqrt{(\Delta x)^2 + (\Delta y)^2}$ as $(\Delta x, \Delta y) \to (0, 0)$.

Proof of Continuity:

Let's rewrite the expression for $f(x_0 + \Delta x, y_0 + \Delta y)$ using $h = \sqrt{(\Delta x)^2 + (\Delta y)^2}$ and observe the following:

$f(x_0 + \Delta x, y_0 + \Delta y) = f(x_0, y_0) + \Delta x f_x(x_0, y_0) + \Delta y f_y(x_0, y_0) + \epsilon(\Delta x, \Delta y)$

Now, express the above equation as:

$f(x_0 + \Delta x, y_0 + \Delta y) - f(x_0, y_0) = \Delta x f_x(x_0, y_0) + \Delta y f_y(x_0, y_0) + \epsilon(\Delta x, \Delta y)$

Taking the limit as $(\Delta x, \Delta y) \to (0, 0)$:

[ \lim_{(\Delta x, \Delta y) \to (0, 0)} \left[ f(x_0 + \Delta x, y_0 + \Delta y) - f(x_0, y_0) \right] = \lim_{(\Delta x, \Delta y) \to (0, 0)} \left[ \Delta x f_x(x_0, y_0) + \Delta y f_y(x_0, y_0) + \epsilon(\Delta x, \Delta y) \right] ]

Since $\Delta x$ and $\Delta y$ both approach 0, the linear terms $\Delta x f_x(x_0, y_0)$ and $\Delta y f_y(x_0, y_0)$ will also approach 0. Moreover, by the property of $\epsilon(\Delta x, \Delta y)$:

$\lim_{(\Delta x, \Delta y) \to (0, 0)} \epsilon(\Delta x, \Delta y) = 0$

Thus:

$\lim_{(\Delta x, \Delta y) \to (0, 0)} \left[ f(x_0 + \Delta x, y_0 + \Delta y) - f(x_0, y_0) \right] = 0$

This implies:

$\lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)$

Hence, $f(x, y)$ is continuous at $(x_0, y_0)$.

Would you like further details or have any questions about this proof?

Here are some related questions:

1. What does it mean for a function to be differentiable at a point?
2. How does differentiability imply continuity in general?
3. Can a function be continuous but not differentiable? Provide an example.
4. What are partial derivatives, and how are they used in multivariable calculus?
5. What is the significance of the function $\epsilon(\Delta x, \Delta y)$ in the context of differentiability?

Tip: When proving the continuity of a function at a point, consider the limit definition of continuity and explore the connection to differentiability, as it often simplifies the proof.