Let $D$ and $E$ be measurable sets and $f$ a function with domain $D\cup E$. We proved that $f$ is measurable on $D \cup E$ if and only if its restrictions to $D$ and $D$ are measurable. Is the same true if measurable is replaced by continuous.
I wrote the question straight out of the book this time to make sure I did this correctly. I mechanically replaced measurable with continuous starting with the measurable after $f$.
Below here is what I wrote before I changed the question I've proven the case where continuous is switched for measurable. I'm just not sure of a meaningful relationship between measurable sets for domains and continuity.