Do we need _strict_ monotonicity for the fixed point lemma for normal functions? I was reading about the fixed-point lemma for normal functions and I was wondering if we could weaken strict monotonicity to non-decreasing. I see why it's necessary for the proof, but is there a continuous non-decreasing ordinal operation without a fixed point? I'm probably missing something obvious but I can't think of any.
You don't need that $f$ is strictly increasing to have at least one fixed point. Consider the sequence $\alpha_n = f^n(0)$. By induction it follows that this is non-decreasing. If $\alpha_n=\alpha_{n+1} $ for some $n$ then we are done as this means that $\alpha_n = f(\alpha_n)$. Else consider the sup of this sequence and follow by continuity that this is a fixed point. 

Of course you cannot get arbitrarily large fixed points as witnessed by the constant $0$ function.
With $\ell(\alpha)=\min\{\,\lambda\mid \lambda\text{ is limit ordinal}, \lambda&gt;\alpha\,\}$, let
$$ f(\alpha)=\begin{cases}\ell(\alpha)&amp;\text{if }\alpha\text{ is limit ordinal}&amp;\\\ell(\ell(\alpha))&amp;\text{otherwise}\end{cases}$$
Then $f$ is continuous, non-decreasing, and has no fixed point.