There is an old Japanese saying, “捨てる神あれば、拾う神あり-Suterukami areba hirou kami ari,” meaning “There’s a god who puts you down as well as a god who picks up you.” In other words, “In this world, some people help you, and some people harm you” or “Fortune and misfortune come alternately.”

For example, when you are fired from an IT company, and then hired by its rival company with a higher salary three months later, your peers will say to you “You're a lucky man. There’s a god who throws you away as well as a god who picks you up.”

I’m curious to know if there are similar sayings in English to “Suterukami areba hirou kami ari.” 


Question: If $f : [-a,a] \rightarrow \mathbb{IR}$ is Scott continuous, why are $f^-$ and $f^+$ measurable?
Answer: First, we note that $\mathbb{IR}$ is ordered by $[a,b] \sqsubseteq [c,d]$ iff $[c,d] \subseteq [a,b]$, which makes $\mathbb{IR}$ into a domain. Now suppose $f: [-a,a]\to \mathbb{IR}$ is Scott-continuous (that is, $f$ preserves all directed suprema).

We want to prove that $f^-$ as defined in the comment above is lower semi-continuous. Let $(x_n)$ be an increasing sequence in $[-a, a]$. So $(x_n) \to s$ where $s = \sup\{x_n:n\in\mathbb{N}\} \in [-a, a]$. We want to show that  $\lim_{n\to\infty} f^-(x_n) = f^-(s)$.

The set $$D:=\{x_n: n\in \mathbb{N}\}$$ is a directed subset of the domain $[-a,a]$ (with the ordering inherited from $\mathbb{R}$), and we have $s=\bigsqcup D$.

For $n\in \mathbb{N}$ we have $f(x_n)\supseteq f(x_{n+1})$ since $f$ is order-preserving (which is implied by $f$ being Scott-continuous). The fact that $f$ is Scott-continuous means that $$[f^-(s), f^+(s)] = f(s)=f(\bigsqcup D) = \bigsqcup f(D) = \bigcap 
\{ [f^-(x_n), f^+(x_n)]:n\in\mathbb{N}\}.$$

So $(f^-(x_n))_{n\in\mathbb{N}}$ is an increasing sequence in $\mathbb{R}$ and it is bounded by $f^+(x_1)$, so it converges. The equations above imply that $\lim_{n\to\infty} f^-(x_n) = f^-(s)$, so $f^-: [-a,a]\to \mathbb{R}$ is lower semi-continuous. A similar argument shows that $f^+$ is upper semi-continuous.