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E54

[29T]Let \(IβŠ‚ ℝ\), \(x_ 0∈ \overline{ℝ}\) accumulation point of \(I\), \(f:Iβ†’ ℝ\) function. Let \(r{\gt}0,tβˆˆβ„,𝜌{\lt}0\); show that

\begin{align*} \limsup _{xβ†’ x_ 0} (f(x) +t) = t+\limsup _{xβ†’ x_ 0}f(x)~ ~ ,~ ~ \limsup _{xβ†’ x_ 0}(r f(x)) = r \limsup _{xβ†’ x_ 0} f(x)~ ~ ,~ ~ \\ \limsup _{xβ†’ x_ 0}(𝜌 f(x)) = 𝜌 \liminf _{xβ†’ x_ 0} f(x)~ ~ . \end{align*}

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Authors: "Mennucci , Andrea C. G." .
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  • limsup, of function
  • liminf, of function
  • real numbers
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