Non-trivial continuous limits of repeated iteration This is not an exercise in any book, I've just thought for a while about this idea and couldn't figure it out, so I thought people at Math.SE might help me.

Define a function transform as follows. For a function $f$ and each x in its domain, define a T-transform $T(f)(x)$ to be the limit of the sequence formed by repeated iteration with $f$ of x $( x, f(x), f(f(x)), f(f(f(x))), ...)$, and if the sequence doesn't converge, define $T(f)(x)$ to be zero. Are the only possible continuous T-transforms of functions from reals to reals constant functions and the identity function? In other words, is there a non-trivial continuous T-transform of some function? I suspect there is none, but I don't know how to prove it.
There are functions $f$ with continuous, nontrivial $T(f)$. Let 
$$
f(x) = \begin{cases}x/2 &amp; x\le 0\\x &amp; 0\le x\le 1\\ (x+1)/2 &amp; x\ge 1\end{cases}
$$
Then 
$$
T(f)(x)= \begin{cases}0&amp; x\le 0\\x &amp; 0\le x\le 1\\ 1 &amp; x\ge 1\end{cases}
$$
Edit: Here is another nice example. Let $g(x)$ be a triangle wave, defined by setting $g(4k+1)=1$ and $g(4k+3)=-1$ for all integers $k$, and interpolating linearly between. Then $g$ satisfies $g(g(x))=g(x)$, implying that $T(g)(x)=g(x)$, so $T(g)$ is the same triangle wave.
Take $f:x\mapsto|x|$ and we will have

$$T(f)(x)=|x|$$

Likewise for $f:x\mapsto-|x|$.