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We are given the function \( f(z) = (x^2 + y) + i(y^2 - x) \), where \( z = x + iy \). To determine the analyticity of \( f(z) \), we need to verify if the Cauchy-Riemann equations are satisfied.

For a function \( f(z) = u(x, y) + iv(x, y) \), where \( u(x, y) \) is the real part and \( v(x, y) \) is the imaginary part, the Cauchy-Riemann equations are:

\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}
\]
\[
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]

Here, the real part \( u(x, y) = x^2 + y \) and the imaginary part \( v(x, y) = y^2 - x \).

Let's calculate the partial derivatives:

1. \( \frac{\partial u}{\partial x} = 2x \)
2. \( \frac{\partial u}{\partial y} = 1 \)
3. \( \frac{\partial v}{\partial x} = -1 \)
4. \( \frac{\partial v}{\partial y} = 2y \)

Now, check the Cauchy-Riemann equations:

- First equation: \( \frac{\partial u}{\partial x} = 2x \) and \( \frac{\partial v}{\partial y} = 2y \). For this to hold, \( 2x = 2y \), which implies \( x = y \).
- Second equation: \( \frac{\partial u}{\partial y} = 1 \) and \( -\frac{\partial v}{\partial x} = 1 \). This equation is always satisfied.

Thus, the Cauchy-Riemann equations hold only along the line \( x = y \). Therefore, \( f(z) \) is analytic only along this line, but not generally analytic everywhere.

From the given options, the correct answer is:
- **f is not analytic at any point.**

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the significance of the Cauchy-Riemann equations in complex analysis?
2. How can we classify singularities of complex functions?
3. What conditions make a function holomorphic?
4. How does conformal mapping relate to analytic functions?
5. Can a function be partially analytic in a region?

**Tip**: Always check both Cauchy-Riemann equations carefully to ensure a function is analytic.

If f(z) = (x^2 + y) + i(y^2 - x), determine if the function is analytic.

This problem examines the function f(z) = (x^2 + y) + i(y^2 - x) to determine where it is analytic by applying the Cauchy-Riemann equations. A detailed solution checks the equations and reveals that the function is not analytic at any point.

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