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E54: [29T]Let $I \subset ℝ$ , $x_{0} \in \overset{―}{ℝ}$ accumulation point of $I$ , $f : I \to ℝ$ function. Let $r > 0, t \in ℝ, 𝜌 < 0$ ; show that

$\begin{array}{r} \underset{x \to x_{0}}{lim sup} (f (x) + t) = t + \underset{x \to x_{0}}{lim sup} f (x), \underset{x \to x_{0}}{lim sup} (r f (x)) = r \underset{x \to x_{0}}{lim sup} f (x), \\ \underset{x \to x_{0}}{lim sup} (𝜌 f (x)) = 𝜌 \underset{x \to x_{0}}{lim inf} f (x) . \end{array}$

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Authors: "Mennucci , Andrea C. G." .

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