EDB — 13R

Exercises

1. [13R]Given \(f:X→ℝ\), define

   \[ f^{*}(x)=f(x)∨ \limsup _{y→ x} f(y) \quad ; \]

   show that \(f^{*}(x)\) is the smallest upper semicontinuous function that is greater than or equal to \(f\) at each point.

   Similarly, define

   \[ f_{*}(x)=f(x)∧ \liminf _{y→ x} f(y) \]

   then \(-(f^{*})=(- f)_{*}\), and therefore \(f_{*}(x)\) is the greatest lower semicontinuous function that is less than or equal to \(f\) at each point.

   Finally, note that \(f^*≥ f_*\).

   Solution 1

   [13S]↺↻