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I want to investigate nested functions; here below are my worked solutions for the integral and derivative of a general nested function.

Let the nested function be defined as

$$y=f(x+f(x+f(x+...))).$$ It can be observed that:

$$f^{-1}y-x=y$$

$$f^{-1}y-y=x$$

$$\frac{\mathrm{dy} }{\mathrm{d} x}(\frac{\mathrm{d} }{\mathrm{d} x}f^{-1}y-1)=1$$

$$\frac{\mathrm{dy} }{\mathrm{d} x}=\frac{1}{(\frac{\mathrm{d} }{\mathrm{d} x}f^{-1}y-1)}$$

Hence,

$$\int y\,dx$$ $$=\int y(\frac{\mathrm{d} }{\mathrm{d} x}f^{-1}y-1)\, dx$$ $$=yf^{-1}y-\frac{y^2}{2}-\int f^{-1}y\,dx$$

Are my workings correct? Also, are there other interesting results that can be obtained from studying such nested functions?

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    It is (the limit of) a function sequence : let $g_0(x) = x$ and $g_{n+1}(x) = f(x+g_n(x))$. The derivative is $g_0'(x) = 1, g_{n+1}'(x) = (1+g_n'(x)) f'(x+g_n(x))$. First of all, you need to prove that it converges – 2017-02-02
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    True, the nested function should be able to converge, such as $\sqrt{x+\sqrt{x+\sqrt{x+...}}}$. Thanks for pointing that out! – 2017-02-02

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