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Well, that's essentially what had to happen with complex analysis, except that there was the real case to generalize from.

But, if you insisted, you could think of mappings. It's actually not so bad because a mapping from R to R is like a path in R, which is very physical. It's the usual way of thinking of the kinematics of moving particles, in fact. Some ideas in calculus may have actually been discovered in such a context. I'm not sure. But people like Newton were aware of that perspective and made use of it. In particular, the fundamental theorem of calculus is probably more intuitive in this picture than in terms of graphs. Adding up velocity vectors gives you position. That's integration. The inverse process is finding velocity vectors from the position as a function of time.

However, the explicit idea of a function as a mapping is surprisingly recent, although it was used implicitly much earlier.

If you mean arbitrary sets A and B with no structure, then there isn't really any calculus there to be done. You at least have to have some sort of algebraic structure there for anything like calculus to even be attempted, and you have to be able to take limits.

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