Solution-verification of function that is lower semicontinuous

Question

Define $f_r(x)=\begin{cases}r\quad x\in V_r\\0\quad o.c. \end{cases}$

$r$ is a rational in $(0,1)$ and $V_r$ is an open set that represents the rational $r$.

I am trying to prove $f_r$ is lower semicontinuous by showing $F_a=\{x\in X:f_r(x)>a\}$ is open for all $a$ real.

Lets define

$F_a(x)=\begin{cases}\varnothing\quad a>0\\V_r\quad r<a\le0\\ X\quad a\le r \end{cases}$

Hence $F_a$ is open. Is my solution correct?

Answer 1

It turns out it is correct indeed