Suppose I have a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ and want to consider the iterates
$$f^{(m)}(x_0) = f(\cdots f(f(x_0)))$$
($m$ times) for some initial point $x_0\in\mathbb{R}^n$. Now assume further that $\lim_{m\rightarrow\infty}f^{(m)}(x_0)=y$ exists. Does it then hold that
$$ f(y) = y$$
even when $f$ is discontinuous? For continuous $f$ this is clear, but my analysis is too rusty to be able to deal with the discontinuous case. Are there standard counter-examples?
It doesn't hold in general. For a discontinuous function, $f(y)$ could be anything at all.
For example: $f(x)=\left\{ \begin{array}{ccc} \frac{x}{2}&\mbox{if}&x\ne 0\\1 &\mbox{if} & x=0\end{array}\right.$
Now look at $x_0=1$