EDB — 13R

Exercises

1. [13R]Given $f:X\to ℝ$, define

   $$f^{\ast }(x)=f(x)\vee \underset{y\to x}{lim sup}f(y);$$

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

   Similarly, define

   $$f_{\ast }(x)=f(x)\wedge \underset{y\to x}{lim inf}f(y)$$

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

   Finally, note that $f^{\ast }\ge f_{\ast }$.

   Solution 1

   [13S]↺↻