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Suppose $f$ is a continuous function defined on $[0,1]$. Let $\operatorname{sgn}(f)$ denote the signal function of $f$. Is $\operatorname{sgn}(f)$ a measurable function?

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up vote 3 down vote accepted

Hint: This functions can take only two values, so it's enough to see that $\{f=0\}$ and $\{f>0\}$ are measurable. These sets are actually respectively closed and open.

(the results holds if $f$ is supposed measurable, not necessarily continuous; in the argument just replace "closed" and "open" by "measurable")

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