My question is motivated by this What is wrong with this funny proof that 2 = 4 using infinite exponentiation? discussion, namely an example of a map $x \mapsto f(x)$ is given, with $$f(x)= (\sqrt{2})^{x}.$$ In that thread, it is stated the map has 2 fixed points, at $x = 2$ and $x=4$. Now to examine their stability we look at the derivative $$f'(x) = (\sqrt{2})^{x} \log \sqrt{2}.$$

Now we have $f'(4) <1$, so $x=4$ should be stable. However, it is claimed, on the contrary, that $x=4$ is, in fact, $\textbf{unstable}$. Why is this?