Oh, these kind of problems happen time and again in real-life maths: you use a lemma until someone points out that you are assuming something which may not take place (like connectedness of the domain, here).

Yea, but like I said, it's a poor example as a "false belief" in this context, because as stated, it isn't false for some unwritten assumptions and is false for other unwritten assumptions. In mathematics, well and actually everything, the importance of assumptions is paramount. The problem is just ill-stated.

This happens in engineering and software all the time. Time and time again, issues being resolved usually revolves around clarifying assumptions.