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}% $$\left\{ x\in X:f\left( x\right) <\lambda \right\} =\left\{ \begin{array}{cl} \emptyset, & \lambda\le 0, \\ X, & \lambda>1 \\ \overline{U}_k, & \frac 1{2^k}<\lambda\le \frac1{2^{k-1}} \end{array} \right. $
we have $f$ is a normal function.
