I was trying to solve this question. [If $f ◦f$ is differentiable, then $f ◦f ◦f$ is differentiable](https://math.stackexchange.com/questions/2603423/if-f-f-is-differentiable-then-f-f-f-is-differentiable). While trying to find the counterexample. I come across the Dirichlet function. $f(x) = \begin{cases} 1 & x \in \mathbb{Q},\\ 0 & x \not\in\mathbb{Q}\end{cases}$. $f ◦f=1$. I landed up on my own definition. I don't know someone might have created this definition before or not.

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> **Definition:-** A function $f:\mathbb R \to \mathbb R$ is $N$-Exhaustible, If $\min\_{n\in \mathbb N} \{\underbrace{f \circ f \circ ...f}\_{\text{n-times}}=Constant\}=N$. 
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According to my definition, Dirichlet function is $2$- Exhaustible. I tried to find $N$-Exhaustible function, where $N>2$. Does there exist any relationship between non-constant $N$-Exhaustible function and differentiability? till now, I got only non-differentiable $N$-Exhaustible functions.Will this definition useful in the future? How do I check whether this definition exists before or not? I have revised the definition. My previous definition was not complete. I hope, My new definition is clear. If not, please help me to correct me.