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The linear function is a pretty good first example: they are very fundamental and everything is simple. But rather than think in terms of the most fundamental thing to come next, I would encourage you to think in terms of typical examples that show the breadth of the concept. Linear function Polynomial of maybe second degree (not linear, but still fairly ...


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The category-theorist in me would argue the most fundamental function is the identity function, so $f(x)=x$ might be a good one to go to next. I say this one because, when you really start diving down into what "fundamental" is, the existence of an identity mapping is really key to a lot of structures. In particular, categories. And, of course, ...


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