TITLE: Is the complex function $f(z) = Re(z)$ differentiable?
QUESTION [10 upvotes]: I am preparing to take my first course in complex variables. I am reading some lecture notes online.  They claim that the function $f(z) = Re(z)$ is continuous but NOT differentiable.  I know the definitions of a limit, of continuity, and of a derivative.  I understand why this function is continuous.  I am trying to show that it is NOT differentiable.

REPLY [2 votes]: Another way to see it, it is that the real part of a complex number can be written with its conjugate: $Re(x) = \frac{1}{2} (x + x^*)$. Since the conjugate function is the classical example of a non-complex-differentiable function (see for exampe this), it follows that the real part is not complex-differentiable.