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Let $f(x)=g(g(g(...g(x))))$, where the function $g$ is applied to $x$ and infinite amount of times. I am assuming that $x$ is real. What is special about points at which $df/dx=0$? Are they related to fixed points?

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  • $\begingroup$ Do you have a good sense for the domain of such a function, let alone where it should be differentiable? For instance, if $g(x) = \sqrt{x}$, then $f$ is real-valued only for $x\geq 0$, and $$f(x) = \begin{cases} 1 & x\neq 0\\0 & x=0\end{cases}$$ and if $g(x) = x^2$, then $$f(x) = \begin{cases} 0 & -1< x <1\\1 & x=\pm 1\\ \infty & \text{otherwise}\end{cases}$$ For a much more interesting example, let $g_\lambda(x) = \lambda x(1-x)$ and study the behavior of $f$ for various values of the parameter $\lambda$. $\endgroup$ – TM Gallagher Apr 25 at 20:20

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