Step Functions in Euclidean Space

Complete the proof of (SC4).

(SC4): We have the following: (a) If $f(\boldsymbol {x})=+\infty$, then $f$ is LSC at $\boldsymbol {x}$ iff $\lim_{\boldsymbol {y} \rightarrow \boldsymbol {x}} f(\boldsymbol {y})=+\infty$. (b) If $f(\boldsymbol {x})=-\infty$, then $f$ is USC at $\boldsymbol {x}$ iff $\lim_{\boldsymbol {y} \rightarrow \boldsymbol {x}} f(\boldsymbol {y})=-\infty$.

Give a formal writeup of the proof of the following [intuitively obvious] assertion: the set of $\boldsymbol {x} \in I$ such that $\boldsymbol {x}$ is a boundary point of some subrectangle of $I$ at some stage is $\lambda_{k}$-null. This assertion was made at the end of the proof of (SC3).

(SC3): If $f(\boldsymbol {x})=-\infty$, then $f$ is LSC at $\boldsymbol {x}$. If $f(\boldsymbol {x})=+\infty$, then $f$ is USC at $\boldsymbol {x}$. [Vacuously true.]