Q2) has a negative answer. Namely, if, e.g., $g(x)=-x$ for all $x\in\mathbb{R}$, then there is no continuous $f:\mathbb{R\rightarrow\mathbb{R}}$ such that $f\circ f=g$.
As to Q3, see, e.g., Theorem 3 in http://yaroslavvb.com/papers/rice-when.pdf.
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Q2) has a negative answer. Namely, if, e.g., $g(x)=-x$ for all $x\in\mathbb{R}$, then there is no continuous $f:\mathbb{R\rightarrow\mathbb{R}}$ such that $f\circ f=g$. |
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