Since $\left\{ x\in X:f\left( x\right) <\lambda \right\} =\left\{ 
\begin{array}{c}
\emptyset,  \\ 
X, \\ 
\overline{U_{k}},
\end{array}
\right. 
\begin{array}{c}
\lambda \leq 0 \\ 
\lambda >1 \\ 
\frac{1}{2^{k}}<\lambda \leq \frac{1}{2^{k-1}}%
\end{array}%
$

we have $f$ is a normal function.