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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?

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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

Check whether the following functions are continuous or not: 1) f(x, y, z) = x^2 + y^2 + z^2, 2) f(x, y) = yx^2 + xy^2, 3) f(x, y) = (x^3 - y^3)/(x + y), 4) f(x, y, z) = tan^{-1}(√x+√y+√z)/(√x+√y+√z)

This problem explores the continuity of multivariable functions including polynomial, rational, and trigonometric functions. Solutions provided with detailed explanations of function behavior and potential discontinuities.

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