I don't think this is a problem about "discontinuity". It's about the monotonicity of the function from income before tax (and insurance, etc.) to income after

I don't think this is a problem about "discontinuity". It's about the monotonicity of the function from income before tax (and insurance, etc.) to income after tax. So I don't think "sharp threshold" is a problem, as long as the law is "if you are making more than this amount, the extra part should be taxed at a higher rate." From other comments I learn that some laws indeed do not work like this. Really?! That's awful.