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$\Rightarrow)$ Continuous images of connected sets are connected. If $f:E\to \{0,1\}$ is continuous, $f(E)\subset \{0,1\}$ is connected so, $f(E)=\{0\}$ or $f(E)=\{1\}$. This means that $f$ is constant.Let $E∈\mathbb{R}^{n}$ be a non-empty subset. Prove that $E$ is connected if and only if any continuous function $f : E→${$0$, $1$} is constant.