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.