Let $f\_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F\_n:= f\_n\circ \dots\circ f\_1$. Then $F\_n$ is continuous. However, the pointwise limit need not be (consider Mateusz's example of
$$
f\_n = \frac{\sqrt{2} x}{\sqrt{1 + 4 x^2}} \qquad F\_n \to \operatorname{sign}(x) 
$$
the sign function).


In general, what can be said about the limits of an iterated function system? Are such functions studied and if so what are they called?


All I figure at the moment is that they must necessary be of [Baire 1](https://en.wikipedia.org/wiki/Baire_function) type.



Infinite composition of continuous functions
Since you’re searching for names: there is a rich literature on *discrete non-autonomous dynamical systems* generated by an infinite family of continuous maps. Here "non-autonomous" means of course that the map you apply depends on the order of iteration, which is exactly your subject.


For instance:


1. Cánovas, J. S. (2006). On ω-limit sets of non-autonomous discrete systems. Journal of Difference Equations and Applications, 12(1), 95-100.
ISO 690
2. Shao, H., Chen, G., & Shi, Y. (2020). Some criteria of chaos in non-autonomous discrete dynamical systems. Journal of Difference Equations and Applications, 26(3), 295-308.
3. Sharma, P., & Raghav, M. (2015). Dynamics of nonautonomous discrete dynamical systems. arXiv preprint arXiv:1512.08868.


You can find a lot more, I suggested those just to make it clear which kind of name/keywords can be useful for you. You'll find lots of results on interval maps, real maps and on more general spaces. It all depends on which specific questions you want to address.


Good news for you: you don't have to consider Baire class 1 functions in their full glory, since given a sequence $\{F\_n\}$ of continuous functions which converges pointwise, the functions $f\_n$ solving the countable set of functional equations $$F\_{n}(x)=f\_n(F\_{n-1}(x)), \ n\ge 2,$$ even when they exist, aren't necessarily continuous. So you're exploring a (topologically small, I guess) portion of Baire class 1 maps.


