okay idk if this will address your confusion but i think there's a real important ontological point here that people elide. in mathematics functions just kinda "exist in platonic reality"; there's no expectation that a function has values that one should be able to compute
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the main thing people will say about this is "functions are not formulas," as in, you're allowed to specify their values via more complicated procedures. so a function f : X -> Y is just some "arbitrary" "assignment" of values, for each x in X, of some f(x) in Y
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but this elides a serious and important point, which is, how are you allowed to specify functions then??? the formal way to do this is to describe them as subsets of the cartesian product X x Y in ZF set theory and it's not very satisfying tbh but that is the formal def
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among other things the question of which functions "exist" is sensitive to the details of set theory which personally i find very unsatisfying. but again idk if this addressed your actual question
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also here's a separate response to thinking about *continuous* functions specifically: continuous functions are, loosely speaking, exactly those functions such that you can approximately know their outputs by approximately knowing their inputs
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this has real epistemic implications. suppose you're trying to do a physics experiment where you're measuring the response of some system to being prodded in some way and you think the response is a function f(x) of the initial conditions. can you "know" this function?
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the problem is that you don't have the power to set up the initial conditions to arbitrary precision. so the best you can hope for to "know" f(x) is to set the initial condition close to x and hope you get something close to f(x). continuity is the condition that this works
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in other words implicitly assuming that various things behave continuously is largely how science can even get off the ground in the first place, and the less true that assumption is (e.g. chaotic dynamics) the more impossible it is to do science even inprinciple
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yeah there are some very relevant concepts here. e.g. humans often behave "discontinuously" 😛 friggin humans eh