Proving f(z)=zRe(z) is Differentiable at Origin

[fredrick08]

Homework Statement

show that the function f(z)=zRe(z) is only differentiable at the origin.

im completely lost with is, its probably very easy.. but don't know how to start.
f(x+iy)=x+iy*Re(z)=x(x+iy)? idk

 

[NT123]

I believe you need to use the Cauchy-Riemann equations, u_x = v_y, v_x = -u_y, where
u is the real part of the function and v is the imaginary part, and u_x is the derivative of u with respect to x etc.

As you stated, the function can be written as x(x+iy) = x^2 + ixy, so u = x^2, v = xy. You will find the functions only satisfy the C-R equations at the origin.

 

[gabbagabbahey]

im completely lost with is, its probably very easy.. but don't know how to start.
f(x+iy)=x+iy*Re(z)=x(x+iy)?

So far so good, now what are the real and imaginary parts, $u(x,y)$ and $v(x,y)$, of this expression? What conditions must the partial derivatives of $u(x,y)$ and $v(x,y)$ satisfy for f(x+iy) to be differentiable?

EDIT: NT123 beat me to it.