Show the given function is not continuous

Question

Let $f:\Bbb C\to \Bbb C$ by $$f(z)=\begin{cases} e^{-1/z^4}\ \text{if}\ z\neq 0\\ 0\ \text{if}\ z=0\end{cases}$$ Show $f$ is not continuous at $0$.

I tired to solve this by finding some appropriate curve so along that curve, its directional derivative is nonzero. Could you help?

Answer 1

Let $z=re^{i\pi/4}$; the limit as $r\to0$ is $$\lim_{r\to0^+}e^{-1/(r^4e^{i\pi})}=\lim_{r\to0^+}e^{1/r^4}=\infty$$ Since $f(0)=0$, the function is not continuous.