Question: Consider ((mathbb{R}, mathscr{B}(mathbb{R}))) and (u: mathbb{R} ightarrow mathbb{R}). Show that ({x} in sigma(u)) for all (x in mathbb{R}) if, and only if, (u) is injective.

Consider $(\mathbb{R}, \mathscr{B}(\mathbb{R}))$ and $u: \mathbb{R} ightarrow \mathbb{R}$. Show that $\{x\} \in \sigma(u)$ for all $x \in \mathbb{R}$ if, and only if, $u$ is injective.

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