Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so require ##c \in dom(f)##. In order to make sense of the limit ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c}##, we must make sure that the expression is a function, on an appropriate domain. It is easy to see that it indeed is, and has domain ##dom{f} - \{c\}##. Thus, we can form the difference quotient function ##\phi: dom(f) - \{c\} \rightarrow \mathbb{R}##, defined by, for ##x \in dom(f) - \{c\}##, ##\phi(x) = \frac{f(x) - f(c)}{x - c}##. So then we would have that ##\lim_{x \to c} \phi(x) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}## as intended. Now, for us to be able to even consider the limit, we would then need ##c## to be a limit point of ##dom{f} - \{c\}##, which we could satisfy by letting ##c## be a limit point of ##dom{f}##. Now, I am aware that one can define differentiation for a function on closed intervals if one considers one-sided derivatives, or one could also go to general open sets which wouldn't cause much more issue since each point in an open set is locally contained in an interval. However, given my flow of reasoning above, should it be correct, wouldn't this mean that one could define differentiation of a point in a domain only for which the point is a limit point? I know that if it's in an interval to begin with, then it is a limit point, but this seems like an arbitrary restriction. Thanks in advance for any response.(adsbygoogle = window.adsbygoogle || []).push({});

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# I Must functions really have interval domains for derivatives?

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