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The question from the image asks, **"Where is the function \( h(x, y) = \arctan\left(\frac{y}{x}\right) \) continuous?"**

### Solution Approach:

The function \( h(x, y) = \arctan\left(\frac{y}{x}\right) \) is a composite function involving the arctangent, which is continuous wherever its argument \( \frac{y}{x} \) is defined.

#### Domain Analysis:
- The arctangent function, \( \arctan(z) \), is continuous for all real values of \( z \).
- However, \( \frac{y}{x} \) introduces a division, meaning the expression is undefined when \( x = 0 \), because division by zero is undefined.

Thus, \( h(x, y) \) will be continuous everywhere except where \( x = 0 \). In other words, the function is continuous on the entire \( \mathbb{R}^2 \) plane except along the line \( x = 0 \), where the function is not defined.

### Conclusion:
The function \( h(x, y) = \arctan\left(\frac{y}{x}\right) \) is continuous everywhere on the \( xy \)-plane, except along the vertical line \( x = 0 \).

Would you like any further details or explanations?

Here are some related questions to explore this concept further:
1. What is the domain of the function \( h(x, y) = \arctan(y/x) \)?
2. How does continuity of composite functions work?
3. What happens to the function \( h(x, y) \) when \( y = 0 \)?
4. What is the significance of discontinuities in multivariable functions?
5. How can the concept of limits help in understanding the continuity of \( h(x, y) \)?

**Tip:** Always check for divisions or square roots in expressions to identify potential points where a function might be discontinuous.

Where is the function h(x, y) = arctan(y/x) continuous?

Explore the continuity of the multivariable function h(x, y) = arctan(y/x). This problem covers concepts such as the domain of rational functions and continuity in multivariable calculus. Suitable for college-level calculus students.

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