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The Fundamental theorem of calculus states:

$f:[a,b]\rightarrow\mathbb{R}$ is integrable and $F:[a,b]\rightarrow\mathbb{R}$ is such that $ \forall x\in\left(a,b\right)$ $F'\left(x\right)=f\left(x\right)$, then $\int_{a}^{b}f=F\left(b\right)-F\left(a\right)$.

Can we rewrite it without using $f$ explicitly. For example,

$F:[a,b]\rightarrow\mathbb{R}$ is differentiable on $[a,b]$, then $\int_{a}^{b}F'=F\left(b\right)-F\left(a\right)$.

If yes, then why mathematician introduced $f$?

If no, could you provide a counterexample that would illustrates the shortcomings of the re-statement.

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  • $\begingroup$ The shortcoming of the re-statement is that it is not the statement we are used to. If you ask me, wriring the $f$ explicitly helps me to understand the theorem. I see no reason why your rewriting is any better than the original. $\endgroup$ – 5xum Jun 15 '15 at 8:42
  • $\begingroup$ If $F'$ isn't continuous or more general a regulated function, taking the Riemann integral may not be defined. $\endgroup$ – abcdef Jun 15 '15 at 8:46
  • $\begingroup$ @abcdef, could you provide an example, please? $\endgroup$ – ji borrob Jun 15 '15 at 8:47
  • $\begingroup$ math.stackexchange.com/questions/293076/… $\endgroup$ – abcdef Jun 15 '15 at 8:58
  • $\begingroup$ Or another example: en.wikipedia.org/wiki/Volterra's_function $\endgroup$ – abcdef Jun 15 '15 at 8:59

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