If a continuous function goes from value A to value B, it must pass through every value in between. In other words, tipping points must necessarily exist.
I propose more specific idea: if you are uniformly uncertain about fractional part of x, then E(round(x+Δx)−round(x))=Δx.
E.g., if you hurry on the way to the subway station without knowing when the next train arrives and got there 10 seconds earlier than if you didn’t hurry, you win exactly the same 10 seconds in expectation.
Yeah, after getting enough people tripped up/upset at me about invoking IVT-like intuitions for discontinuous functions[1], I suspect something like the above is the subtler point I should’ve led with. Elsewhere I wrote I think part of the argument is that if you have a complicated distribution between a bunch of unknown discontinuous functions “in reality”, from your epistemic state, it would often essentially look continuous to you when you combine the probabilities together, and you should treat them as such.
I think your formalism is helpful/might aid in thinking more clearly, but I’m also worried people would jump at it if their uncertainty is slightly non-uniform (without noticing that changing the math a tiny bit only changes the endline result a tiny bit).
In a lot of situations you can still treat your situation as locally linear despite non-global uniformity (see my third point on differentiable functions being locally linear), but that argument is more about “negotiating price,” my first (IVT-inspired) point was establishing that it’s possible to have an effect at all.
Total agree with the train example being a clear elucidation. I’ve used it before in other contexts when trying to explain EV-style reasoning more directly.
Obviously IVT doesn’t hold for all discontinuous functions. But IVT-style intuitions still hold up for reasons like the ones you illustrate, most of the time.
I propose more specific idea: if you are uniformly uncertain about fractional part of x, then E(round(x+Δx)−round(x))=Δx.
E.g., if you hurry on the way to the subway station without knowing when the next train arrives and got there 10 seconds earlier than if you didn’t hurry, you win exactly the same 10 seconds in expectation.
Yeah, after getting enough people tripped up/upset at me about invoking IVT-like intuitions for discontinuous functions[1], I suspect something like the above is the subtler point I should’ve led with. Elsewhere I wrote I think part of the argument is that if you have a complicated distribution between a bunch of unknown discontinuous functions “in reality”, from your epistemic state, it would often essentially look continuous to you when you combine the probabilities together, and you should treat them as such.
I think your formalism is helpful/might aid in thinking more clearly, but I’m also worried people would jump at it if their uncertainty is slightly non-uniform (without noticing that changing the math a tiny bit only changes the endline result a tiny bit).
In a lot of situations you can still treat your situation as locally linear despite non-global uniformity (see my third point on differentiable functions being locally linear), but that argument is more about “negotiating price,” my first (IVT-inspired) point was establishing that it’s possible to have an effect at all.
Total agree with the train example being a clear elucidation. I’ve used it before in other contexts when trying to explain EV-style reasoning more directly.
Obviously IVT doesn’t hold for all discontinuous functions. But IVT-style intuitions still hold up for reasons like the ones you illustrate, most of the time.