- #1

saraaaahhhhhh

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## Homework Statement

Show that f(z) = zRez is differentiable only at z=0,

find f'(0)

## The Attempt at a Solution

This should be easy. I find the limit as z_0 approaches 0 of [f(z+z_0) - f(z)]/(z_0) for this function...expand it out, simplify, and find what the limit is when z_0 is purely imaginary vs purely real. I did this, and I got x for the real part and x_0 + 2x + iy for the imaginary part. They're different, so it's not differentiable everywhere.

But apparently it's differentiable only at z=0. So I tried plugging in z=0 and resolving, and I got different values for z_0 real and imaginary.

I get, for the limit when z_0 is only real: 1

When z_0 only imaginary: x_0.

What's going on? Is there some other way to differentiate a complex function that's not using the limit?