No you don't. Earning more never causes you to take hom eless after taxes (at least in any country I know of). That's a common fallacy about them amongst people who don't understand the tax code.
Tax rates are marginal. So for instance if the tax rate was 20% up to $100k, then 30% afterward, you'd pay 20% on your first $100k, then 30% from $100,001 and up.
Income tax is probably always marginal, but in the UK at least this kind of thinking does apply to selling houses - when a house is sold in the UK the buyer must pay "stamp duty", which is a percentage of the value of the house, with the percentage changing at certain thresholds. For example at £250,000 the stamp duty changes from 1% to 3% - but the percentage applies to the entire value of the house. So if you sell for 250,000 the buyer pays 250000 * 0.03=£7500, compared to 249999 * 0.01=£2,499.
That introduces some distortions into house pricing, making it very difficult to sell a house in the £250,000 to £265,000 range - because you are competing with houses just under the threshold which work out significantly cheaper for the buyer. As someone selling a house which falls into that price range, you might be better off asking for slightly less from the sale (versus asking for just over the threshold and waiting much longer to find a buyer).
Hmm. You are right. All of the examples I was taught with used fictional examples of magical taxes. I though surely there must be somewhere... but no, just a clever fiction, repeated ad absurdum to make word problems for 5th graders and then glossed over by time until I took it for reality. Ouch. Harder still because with a moments reflection I should have seen its absurdity(1).
But the step function thing was the point, not the taxes. I'll put in an edit and get that last bit ignored.
(1) Be careful what you take as truth when young and don't rethink later.
Tax rates are marginal. So for instance if the tax rate was 20% up to $100k, then 30% afterward, you'd pay 20% on your first $100k, then 30% from $100,001 and up.