complex analysis -bounded but not holomorphic function What are some examples of functions that are bounded but not holomorphic?
I know that some the not holomorphic functions are
Absolute value function, Piecewise functions (not analytic where the pieces meet)
 A: Well, you can take any bounded discontinuous function. For example,
$f(z) = 1$ if $z = 0$ and $f(z) = 0$ otherwise is bounded and discontinuous and hence not holomorphic.
A: In fact, it is the case that almost all bounded functions are nonholomorphic, by which I mean that if you pick a random bounded function on some domain $D$, there is a $0\%$ chance it will be holomorphic.  Intuitively, one can simply assign all the elements of $D$ to arbitrary meaningless values (as long as the values are bounded) and the result will be discontinuous, and thus nonholomorphic.
A: You can start with the absolute value function and make it bounded, like for example $$
  f(z) = \frac{|z|}{1+|z|} \leq 1 \text{.}
$$
You get (assuming that $z=u+iv$ and using that $|z|=\sqrt{u^2+v^2}$) $$\begin{aligned}
  \frac{\partial \Re f}{\partial u} &= \frac{-u|z|^{-1}(1+|z|) + u}{(1+|z|)^2} = -\frac{u}{\sqrt{u^2+v^2}\left(1+\sqrt{u^2+v^2}\right)^2} \\
  \neq\frac{\partial \Im f}{\partial v} &= 0 \text{.}
\end{aligned}
$$
i.e. $f$ is not holomorphic.
