Suppose $X$ is a metrizable space. A real-valued function $f:X \rightarrow \mathbb{R}$ is upper semicontinous if for any real number $c$, its preimage $f^{-1}(-\infty,c)$ is open in $X$.
In this post, we see that every lower semicontinuous can be expressed as thesupremum of increasing continuous functions, where the sequence of continuous functions is defined as $f_k(x) = \inf \{ f(y) + k d(x,y): y \in X \}$.
I am curious as to how would one show that an upper semicontinuous function $f$ can be expressed as the infimum of non-increasing continuous functions that converge pointwise, by using the definition of upper semicontinuity above, which is $f^{-1}(-\infty,c).$
Best Answer
If $f$ is usc. then $-f$ is lsc.
Let $\phi_k(x) = \inf_y (-f(y) + k d(x,y)) $, and $f_k(x) = - \phi_k(x) = -\inf_y (-f(y) + k d(x,y))$.
Rearranging gives $f_k(x) = \sup_y (f(y) - k d(x,y))$.