Discussion
Which one of the following functions is differentiable at \(z = 0\) but NOT differentiable at any other point in the complex plane \(\mathbb{C}\)?
Answer: A
The function is \(f(z) = z|z|\).
At \(z = 0\) the difference quotient is \(\frac{f(h) - f(0)}{h} = |h| \to 0\), so \(f'(0) = 0\) exists. Elsewhere write \(r = \sqrt{x^{2} + y^{2}}\), \(u = xr\), \(v = yr\). Then
\[ \begin{aligned} u_{x} &= r + \tfrac{x^{2}}{r}, \quad v_{y} = r + \tfrac{y^{2}}{r} \\ u_{y} &= \tfrac{xy}{r}, \quad v_{x} = \tfrac{xy}{r} \end{aligned} \]
The Cauchy-Riemann equations \(u_{x} = v_{y}\) and \(u_{y} = -v_{x}\) need \(x^{2} = y^{2}\) and \(xy = 0\), which together force \(x = y = 0\). So \(z|z|\) is differentiable only at the origin.
\(\sin z\) and \(e^{-z^{2}}\) are entire, differentiable everywhere. \(e^{1/z}\) is differentiable at every \(z \ne 0\) but has an essential singularity at 0, so it is not even continuous there: the reverse of what is asked.
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