wonderings

consider some function that is zero at every point. now set the output of some input x₀ as f(x₀), where f is some differentiable function (say, f(x) = x²), so that the function is zero everywhere except at that point. now set some other point x₁ to equal f(x₁). repeat this process uncountably infinitely many times, until this new function equals f at every point.

is this function continuous and differentiable?

it seems like it's not, because at every point in its creation, it's just a string of points. then again, it seems like it is, because it equals f at every point, and f is both continuous and differentiable. then again, maybe making a function like this is impossible, because you have to do uncountably many definitions in order to cover the reals, and the whole point of uncountable numbers is that even with infinite time, you can't do uncountably many things.

but if you can't construct a function like this, how can you construct any function? what does continuity even mean, anyway? now that i think about it, the ideas of limits and continuity seem even more shaky than the infinitesimals they were meant to replace. they make sense, but they don't seem like they could ever be rigorously justified.