Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 1 / 11D (Analysis I)

Fix $x\in (0,1)$ and put $x_0=x$, $x_{n+1}=f(x_n)$. Then $0<x_{n+1}<x_n\le x_0<1$. The bounded monotone sequence theorem gives a limit $\ell \in [0,x_0]$. If $\ell >0$, it lies in the domain, where continuity allows passage to the limit in the recurrence: $\ell =f(\ell )$. That is a fixed point, contrary to $f(\ell )<\ell$. Hence $\boxed{f^n(x)\to 0\text{ for every }x\in (0,1)}$. Continuity is used at a possible positive limiting point, not at the excluded endpoint zero.

No: pointwise convergence need not be uniform convergence. The continuous function $f(x)=x^2$ maps $(0,1)$ into itself and is strictly below the identity. Its iteration of a map givesFor every finite $n$, these values tend to $1$ as $x\uparrow 1$, so ${\sup }_{0<x<1}{f}^n(x)=1$. With $\epsilon =1/2$, choosing $x>2^{-1/2^n}$ gives $f^n(x)>1/2$. Thus no single iterate works for every starting point, although each fixed starting point tends to zero.

Use the discontinuous exampleIn both pieces $0<f(x)<x$. For $x>1/2$, the image stays strictly above $1/2$, and induction givesThis positive limiting point is not a fixed point, since $f(1/2)=1/4$. The discontinuity is precisely why the limit cannot be passed through $f$. Therefore the pointwise zero-limit conclusion can fail without continuity.

No: there need not be even one orbit tending to zero. Partition the domain into the disjoint intervalsand, for the unique $j$ with $x\in I_j$, defineThe left endpoint is strictly below $x$, so $0<f(x)<x$. Moreover $f(x)>1/(j+1)$ and $f(x)\le x\le 1/j$, so $f(x)$ remains in the same interval. For every iterate,Every starting point belongs to some finite-index interval, so this proves the claim for all $x\in (0,1)$. The open left endpoints matter: an orbit approaches a boundary but never crosses it. This is discontinuous trapping of decreasing iterates, not a violation of the bounded monotone sequence theorem; each orbit does converge, just to a positive value where continuity fails.