$g(x)=\sup \{f(y): y\in B(x)\}$ is lsc on $R^{n}$ where $B(x)$ is a open ball with fixed radius $r$
Solution 1:
If $d(y,x_0) <r$ then $d(y,x_n) <r$ for $n$ sufficiently large. [To be explicit $d(y,x_n) \leq d(y,x_0)+d(x_0,x_n) <r$ whenever $d(x_n,x_0) <r-d(y,x_0)$]. Fix one such $n$. Then $y \in B(x_n)$ so $f(y) \leq a$. This is true for all $y \in B(x_0)$ . Taking supremum over $y$ we get $g(x_0) \leq a$.