Suppose, e.g., that you can get $5k/yr in benefits if you make less than $10k/yr in other revenue and $0 otherwise. Unless you have a viable strategy for pushing past $15k/yr it's a strong financial disencentive against actually working, and even then your incremental ROI isn't very good past that cliff (if it takes an extra hundred hours to push to $15.1k/yr, then compared to your $10k/yr option you're only making $1/hr for the extra work).
This doesn't sound monotonic. This sounds like a mapping from pre-benefit income to post-benefit income which sends just under $10k/yr to just under $15k/yr, but sends just over $10k/yr to just over $10k/yr. So it sends a larger input to a smaller output.
I think I see the definitions we're disagreeing on. I'll lay out what I meant and then address your thing.
1. A couple levels up, the function somebody requested being continuously differentiable was the "benefits." You seem to be looking at the total post-benefit income instead.
2. It's not totally clear, but you _might_ also be using "monotonic" to refer to "monotonic increasing" or "strictly monotonic increasing" instead (the total income function in my example isn't even monotonic, so this is just me reading between the lines in the wording of your reply).
The cliff issue still exists though (and still exists in the differentiable version -- you want bounds on the derivative ideally to prevent the cliff issue, and for unrelated reasons you probably want other properties like the benefits not increasing as a function of pre-benefit income). Suppose you have a strictly monotonic increasing function mapping pre to post benefit income. That function can still, e.g., have a long region with a high slope, followed by a long region with a low slope, and some curvy thing connecting those. It's still continuously differentiable and strictly monotonic increasing, but the incremental value of work in the low-slope region is low (by definition). You might make a dollar in pre-benefit income and $0.10 in post-benefit income, so at minimum wage you're back to a <$1/hr situation unless you can make enough money to push substantially past the right side of that low-slope region (which we're assuming exists, else it basically says you have a 90% tax rate if you make too much money, and in-context "too much" would be near poverty levels, so even people wanting that sort of thing for the ultra rich wouldn't think that to be a good idea).
I'll go further and say what we probably want is for the derivative of net income as a function of earned income to be monotonic increasing but max out less than 1. So that there aren't ranges of income where you are receiving very little per dollar earned and then after some point start receiving more per dollar.
But solving benefit cliffs really just means having earned=>net income strictly increasing with the marginal rate reasonable, say at least 30 cents more net income per earned income. Under that constraint, you could have ranged where net income grows slower until you hit some higher dollar amount of earnings, but imo that should also not be desirable.
Suppose, e.g., that you can get $5k/yr in benefits if you make less than $10k/yr in other revenue and $0 otherwise. Unless you have a viable strategy for pushing past $15k/yr it's a strong financial disencentive against actually working, and even then your incremental ROI isn't very good past that cliff (if it takes an extra hundred hours to push to $15.1k/yr, then compared to your $10k/yr option you're only making $1/hr for the extra work).