We always speak of the derivative as being the “best linear approximation”. And we also speak of linearizing. However, what does this really mean? For a given function $F$, what conditions on it make the claim “the derivative is the best linear approximation to $F$” true?
Are there functions that can’t be “locally linear” or locally approximated? If so, are these just mostly pathological, and we don’t have any interest in them (e.g. they don’t really show up in math)?
Are there important functions or mathematical objects that don’t really subject themselves well to the tools of analysis and approximation? (I understand this is a very broad and vague question.) I mean, there may be mathematical objects that we don’t know whether they are amenable to such efforts, but are there (important) objects where we are sure they definitely aren’t? Sort of like how abstract algebra/Galois theory showed the limitations of using radicals, giving rise to the notion of unsolvability?