In "A domain theoretic account of Picard's theorem" (http://www.doc.ic.ac.uk/~dirk/Publications/icalp2004.pdf), the authors assert the following.

Let $\mathbb{IR}$ be the interval domain $\lbrace [a^-,a⁺] | a^- \leq a^+, a^-, a^+ \in \mathbb{R} \rbrace \cup \lbrace \mathbb{R} \rbrace$.

Suppose $f = [f^-,f^+] : [-a,a] \rightarrow \mathbb{IR}$ is Scott continuous. Then $f^-$ and $f^+$ are measurable.

Their proof consists in asserting that $f⁻$ and $f⁺$ are lower (resp. upper) semi-continuous, which I have tried without success to establish. Can anyone help with either an explanation or a reference that proves it?


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