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A continuous bracket could be defined by a single equation. You'd plug in your income and you'd get out your taxation. No need to look up what bracket you are in.
If you expect the average person to be able to understand an algebraic equation better than the existing system, then I’d suggest you get out of your social bubble and meet more ‘average’ people.
I honestly think they'd understand it about as much as they'd currently understand it - that is, not understanding it very much at all. So I think it would be about the same level of understanding but it would make a lot more sense and would be easier to calculate with.
It'd also remove a lot of incentives for squeezing your income/pension contribution to align with certain thresholds (i.e. tax bracket thresholds).
Give it a shot. Let me see an equation.
For example, a sigmoid function (click link for equation). You'd need to mess with the constants to align the function with a range of incomes but the general shape will be the same - a low, almost-zero taxation rate for those who earn the least, rising to a threshold (perhaps even 100%, but a lower value like 75% would probably work as well), giving a high taxation to those who earn the most.
You think the general population is going to have an easier time understanding the sigmoid function that some simple multiplication and addition?