How to prove that there exists no continuous function $f:[0,1] \to \{A \in M_2(\mathbb{R})|A^2=A\}$ such that $f(0)=0$ and $f(1)=I$.
2026-03-28 00:28:54.1774657734
Continuous function from $[0,1]$ to set of idempotent matrices in $M_2(\mathbb{R})$
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Consider $g : \{ A \in M_2(\mathbb{R})| A^2=A\} \to \{0,1\}$ given by $g(A)=detA$. Then $g\circ f$ is a continuous function from $[0,1] \to \{0,1\}$ . Then $g(0)=0$ and $g(1)=1$. So, $g$ is surjective. Now $[0,1]$ is connected but $\{0,1\}$ is not connected. Hence such f does not exist.