With respect to the countries about which I am familiar, income tax is partitioned into tax "brackets", meaning that the rate of taxation varies discretely. Different tax rates are therefore applied to different "blocks" of an individual's annual income. The UK, for example, also has a personal allowance - a pot of money an individual can earn tax-free.
This system seems counter-intuitive to me - it complicates annual tax calculation, as income needs to first be partitioned. Why not apply a continuous and smooth (but obviously non-linear) scale to taxation. The rate would approach a limit and start slow, forming a kind of sigmoid-shaped distribution. The tax rate could then be applied in one motion and could be arguably more "fair."
I'm editing this because it seems to have received many responses - which I didn't anticipate, so I'll offer some of my reasoning behind the question.
In my eyes, and from going through some of the comments & answers which skewed my opinion slightly, I see the following pros and cons:
The commonly held perspective of "There's no point in taking that job for $N/annum because I'd be taxed through the roof." at some salary thresholds would be partially diminished.
There's no two ways about it - it would give finer-grained control over income taxation at the expense of ease-of-understanding. It could offer a fairer tax system, in spite of public impression of complexity.
Tax is digital in many countries - we shouldn't have to be narrow-minded about the possibilities for tax any more. Tax tables are (or at least should be) a thing of the past.
Personal finance has hitherto been blessed with ease of calculation, even with pen and paper. This would throw a spanner in the works.
Would there be a limit to how complex the function could become? Could governments exploit the system by making the function so convoluted or employ devious mathematical techniques (e.g. around the rounding of pennies) as to extort certain parts of the populous?
The system cannot be changed to accommodate one group without affecting all, however marginally. (Actually, this isn't completely true if a piecewise function is adopted, but that's going even further down the rabbit hole.) This would seriously impact political campaigns for example, as intention can not be so easily conveyed.
The fragile tax legal framework would be seriously threatened by such a major change.