To show that the function $f(z) = z |z|$ is differentiable at $z = 0$, we need to check whether the definition of the derivative holds at this point.

Definition of Differentiability

The function $f(z)$ is differentiable at $z = 0$ if the following limit exists:

$\lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}$

Step 1: Calculate $f(0)$

First, we need to evaluate $f(0)$.

$f(0) = 0 \cdot |0| = 0$

Step 2: Expression for the Difference Quotient

Next, we need to compute the difference quotient:

$\frac{f(h) - f(0)}{h} = \frac{f(h)}{h} = \frac{h |h|}{h} = |h|$

Thus, the difference quotient becomes:

$\frac{f(h) - f(0)}{h} = |h|$

Step 3: Take the Limit as $h \to 0$

Now, we take the limit of the difference quotient as $h \to 0$:

$\lim_{h \to 0} |h| = 0$

Step 4: Conclusion

Since the limit exists and equals 0, we conclude that:

$\lim_{h \to 0} \frac{f(h) - f(0)}{h} = 0$

Therefore, the function $f(z) = z |z|$ is differentiable at $z = 0$, and its derivative at this point is $f'(0) = 0$.

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Would you like a more detailed explanation of any part of the proof, or do you have additional questions? Here are some related questions you might find interesting:

1. What does it mean for a function to be differentiable at a point in the complex plane?
2. How does the behavior of $f(z) = z |z|$ differ from differentiability in the real case?
3. What is the relationship between the modulus $|z|$ and the differentiability of complex functions?
4. Can the function $f(z) = z |z|$ be extended to be differentiable at all points in the complex plane?
5. What is the meaning of the complex derivative at points where the function is not differentiable?

Tip: Differentiability in the complex plane requires the limit of the difference quotient to exist and be the same regardless of the direction from which $h$ approaches 0.