The following function
$$f(x)=-2 \cos \left(\sqrt{2} \arccos \left(\frac{x-1}{2}\right)\right)+1$$
has interesting property to form a continuous curve with its own integer iterations. The following image illustrates this property:
![alt text][1x] [<sup>(source)</sup>][1]
Here blue is $f(x)$, red is $f(f(x))$, yellow is $f(f(f(x)))$ and green is $f(f(f(f(x))))$. It seems that all these functions form a continuous, and, probably, smooth curve.
The question is what is the general criterion for a function to have such property. Can you point some more examples of functions with such property?
P.S. If to use the following function
$$f(x)=-2 \cos \left(\sqrt{2} \arccos \left(\frac{x-1}{2}\right)\right)+1.1$$
the curve becomes as below with one continuous branch and numerous closed circular branches.
![alt text][2x] [<sup>(source)</sup>][2]
For function
$$f(x)=-2 \cos \left(\sqrt{2} \arccos \left(\frac{x-1}{2}\right)\right)+0.9$$
the curve is completely continuous and seems to be smooth.
![alt text][3x] [<sup>(source)</sup>][3]
[1]: http://static.itmages.ru/i/10/1201/h_1291177404_9062e5977d.png
[2]: http://static.itmages.ru/i/10/1201/h_1291214023_dcadf36911.png
[3]: http://static.itmages.ru/i/10/1201/h_1291213783_96931fa742.png
[1x]: https://i.sstatic.net/81zbL.png
[2x]: https://i.sstatic.net/R2qMo.png
[3x]: https://i.sstatic.net/8IV1X.png