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In my school calculus textbook, the derivative is introduced by saying "Let $f(x)$ be a real valued function defined on an open interval $(a,b)$ and let $c$ belongs to $(a,b)$. Then $f(x)$ is to be differential at $x = c$ , iff the limit of $f(x) - f(c) / (x - c) $ exists finitely." My question is why it is always mentioned 'open set'. Is not differentiation applicable on a point which belongs to a closed set? Please help.

M47145
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  • http://math.stackexchange.com/questions/161910/most-general-a-subseteq-mathbb-r-to-define-derivative-of-f-a-to-mathbb-r – Siminore Apr 28 '14 at 11:23

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First, note there are sets that are not open nor closed. Then, what happens if your closed set consists of a single point?

The key here is that in order for the definition of differentiability at $x$ to make sense, you need the function to be defined in a neighborhood of $x$. So you need $x$ in the interior of the domain of definition of $f$.

There are generalization to limit cases, for instance when the interval of dedinition is of the type $[x,c)$ or $(c,x]$. In this case you may define the limit from right and left, respectively, and you can define the right- and left-derivative of $f$ at such an $x$.

user126154
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  • sir can u elaborate with an example? what is dedinition? –  Apr 28 '14 at 10:41
  • the definition is that you given. Definition: $f$ is differetiable at $x = c$, iff $\lim_{x\to c} (f(x) - f(c))/(x - c)$ exists and it is finte. – user126154 Apr 29 '14 at 13:23