I think you have misconceptions about what continuity means and I’m not even sure what you mean with holes. As a simpler example: The map that multiplies each

I think you have misconceptions about what continuity means and I’m not even sure what you mean with holes. As a simpler example: The map that multiplies each rational number by -1 and keeps irrational numbers as they are is bijective (it’s its own inverse), however it is nowhere continuous.