Let's analyze the continuity of each function step by step:

1) $f(x, y, z) = x^2 + y^2 + z^2$

This is a polynomial function in three variables. Polynomial functions are continuous everywhere, as they do not have any points of discontinuity. Therefore:

• Conclusion: $f(x, y, z) = x^2 + y^2 + z^2$ is continuous for all real values of $x, y, z$.

2) $f(x, y) = yx^2 + xy^2$

This is a polynomial in two variables $x$ and $y$, and polynomial functions are continuous everywhere, regardless of the values of $x$ and $y$.

• Conclusion: $f(x, y) = yx^2 + xy^2$ is continuous for all real values of $x$ and $y$.

3) $f(x, y) = \frac{x^3 - y^3}{x + y}$

Here we have a rational function. Rational functions are continuous everywhere except where the denominator is zero. In this case, the function may have a discontinuity where $x + y = 0$, which simplifies to $y = -x$.

To analyze the potential discontinuity, let's check the behavior of the function as $y \to -x$. We use factorization:

$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$

Thus, the function becomes:

$f(x, y) = \frac{(x - y)(x^2 + xy + y^2)}{x + y}$

At $y = -x$, this function simplifies by canceling out the common factors. Therefore, we can conclude that there is no actual discontinuity, and the function is continuous at all points, including $y = -x$.

• Conclusion: $f(x, y) = \frac{x^3 - y^3}{x + y}$ is continuous everywhere.

4) $f(x, y, z) = \tan^{-1}\left( \frac{\sqrt{x} + \sqrt{y} + \sqrt{z}}{\sqrt{x} + \sqrt{y} + \sqrt{z}} \right)$

This function simplifies to:

$f(x, y, z) = \tan^{-1}(1)$

Since $\tan^{-1}(1)$ is a constant function (equal to $\frac{\pi}{4}$) and constant functions are continuous everywhere, this function is continuous for all real values of $x$, $y$, and $z$.

• Conclusion: $f(x, y, z) = \tan^{-1}(1)$ is continuous everywhere.

Summary

1. $f(x, y, z) = x^2 + y^2 + z^2$ is continuous.
2. $f(x, y) = yx^2 + xy^2$ is continuous.
3. $f(x, y) = \frac{x^3 - y^3}{x + y}$ is continuous.
4. $f(x, y, z) = \tan^{-1}(1)$ is continuous.

Would you like further details or explanations on any specific function?

Here are 5 related questions:

1. How do you check for continuity of a function at a specific point?
2. What types of functions are always continuous?
3. How can we handle removable discontinuities in a function?
4. What is the difference between pointwise and uniform continuity?
5. How do multivariable limits relate to the continuity of functions?

Tip: Remember that polynomial functions are always continuous, and rational functions are continuous wherever their denominators are non-zero