They should be. And for the great majority of citizens, they should be calculated automatically along with taxes without the need for filing/paperwork. But we must think of the Intuit shareholders and the harm that that would cause them.

There is idea behind that, but continuous is not enough.

The variable is all transfers, taxes and benefits T = [all taxes - all benefits] as function of income per person (including children). T starts negative (benefits are negative taxes).

Goal: monotonously increasing effective marginal T rate.

To calculate the amount of your childcare benefit take the negative of the your adjusted gross income (box 26a), divide it by 10,000, and add 2. Then take the arctan of that result and multiply by 5,000. Add 2500π and write the result in box 46. You may round the value of π to 2 digits. If you are filing as head of household, divide the amount in box 46 by the definite integral from 0 to your adjusted gross income (box 26a) divided by 10,000 of sin^2(x)/(x^2 + 1) dx, and write that amount in box 48. If you are unable to find the function of the anti-derivative, IRS rules allow you to approximate with a Riemann sum using the midpoint rule and a rectangle width of 0.1.

I would guess that was not a joke. The benefits could drop all the way to 0 at a reasonable point unless you impose the stronger condition that the function be analytic.

You jest, but as an unemployed student I was approved for $6/mo of SNAP.

Like... for what possible purpose would the calculation even go that low? Just deny the application instead of wasting everyone's time and resources. I have no idea how long they put $6 into the account. I never bothered to look.

Why continuously differentiable and not just monotonic?

That can still have work disincentives; anywhere the magnitude of slope of the benefits is close to (or steeper than) the slope of the income as it phases out, then working more can get you no gain (or lose you money).

Continuously differentiable doesn't mean your SNAP check couldn't go to zero.

It's not sufficient to make cliffs impossible, you just never want the effective marginal tax rate go above a certain threshold.

It's insane that we cap marginal tax rates for the wealthy below 50% “because they need incentives to work harder” yet the working class family is facing an effective marginal rate near 100% because of reduced benefits.

The solution would simply be to stop making benefits decrease when salary goes up.

“But it's going to be insanely expensive” one may say, but it's an accounting illusion. All we need to do to break the illusion would be to stop counting gross public spendings and taxes and instead count the net public spendings/taxes for each individual (that is, over the whole population you take the difference between what they pay and what they receive and that gives you how much they contribute or how much they cost, instead of the current accounting system where we count people paying for their own benefits).

What's really expensive is the economic inefficiency of the current system.

If you continue with the idea of transfers, which is, in effect, the current government buying off parts of its constituents, then "democracy", whatever has remained of it in the West, will die for good. What happens if I receive money from the Government but in the same time I'm also actively opposed to said Government's actions? Will I be allowed to speak against the Government that is, as a matter of fact, paying me? Will I have second thoughts of doing it? Will the Government cut off aid to me if I'm too vocal against said Government's actions?

All this to say that all we're doing is turning most of our countries' citizens into de facto slaves, people with no political free-will and who are well-aware that if they were to speak out against the powers that be they risk destitution.