Show $f(z) = |z|^2$ is differentiable only at $z = 0$

Question

So far I have rewritten the following expression:

$$f'(z) = \lim_{\Delta z\rightarrow0}\frac{f(z + \Delta{z})-f(z)}{\Delta{z}}$$

As such:

$$f'(z) =\lim_{\Delta z\rightarrow 0} (\bar{z} + \bar{\Delta{z}} + z\frac{\bar{\Delta{z}}}{\Delta{z}}) $$

This limit only exists if each of the three terms in the summation have limits. Clearly the first two do, so I am currently evaluating the third term, that is, seeing if.. $$z(\lim_{\Delta{z}\rightarrow 0}\frac{\bar{\Delta{z}}}{\Delta{z}})$$

..exists for $z=0$. I don't think the limit above exists: if we allow $\Delta{z}$ to approach $0$ through the real numbers, then the numerator and denominator are the same and the limit evaluates to 1. Else, if we let $\Delta{z}$ approach $0$ from 'above' (through the imaginary numbers), we have the denominator is the negation of the numerator and so the limit is $-1$. Hence the limit is undefined.

However, would $0$ times a limit that does not exist be $0$? I'm just not sure how to prove this term does indeed evaluate to some value instead of being undefined; it's clear that for $z \neq 0$ that the expression is undefined, but for $z=0$ I'm confused if $0$ times an undefined limit is $0$?

Answer 1

Think of a limit in $\mathbb C$ as a limit in $\mathbb R^2$. You can show a limit does not exist by taking two different paths.

Think of $\Delta z$ as $\Delta x +i \Delta y$ (where $\Delta x$ and $\Delta y$ are real), and take the limit for $(\Delta x, \Delta y)\to(0,0)$ along several curves. For instance: $$\lim_{(\Delta x,\Delta y)\to (0,0)} \frac{\overline {\Delta x+i \Delta y}}{\Delta x+i \Delta y},$$ becomes for $\Delta x=0$ $$\lim_{\Delta y\to 0} \frac{0-i \Delta y}{0+i \Delta y}=-1$$ and for $\Delta y=0$ $$\lim_{\Delta x\to 0} \frac{\Delta x-i \cdot 0}{\Delta x+i \cdot 0}=1.$$ This proves that the limit does not exist.

In any case, that doesn't let you conclude anything with your calculations. I suggest that you go back to the beginning and apply the same idea to the whole limit.

You should get that the limit does not exist if $z\neq 0$, that is if $z=x+iy$ with $(x,y)\neq (0,0)$. For the case $(x,y)=(0,0)$ it is not enough in general to see that both directions give the same limit, so you should use the definition.

However, it is possible that you have read or heard about Cauchy Riemann equations. The theorem states:

If $f(x+iy)=u(x,y)+i\cdot v(x,y)$, with $u, v\colon \mathbb R^2 \to \mathbb R$ two differentiable functions, then $f$ is differentiable at $x_0+iy_0$ IF AND ONLY IF $$\frac {\partial u}{\partial x}(x_0,y_0)=\frac {\partial v}{\partial y}(x_0,y_0)$$ and $$\frac {\partial u}{\partial y}(x_0,y_0)=-\frac {\partial x}{\partial y}(x_0,y_0).$$

Try using this theorem. Since for $z=x+iy$, $$|z|^2=|x+iy|^2=x^2+y^2=(x^2+y^2)+0i,$$ here we have $u(x,y)=x^2+y^2$ and $v(x,y)=0$.