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Are there any cases which the First Fundamental Theorem of Calculus would fail?

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    General idea is good, but too many words, hiding essential idea. Let $f(x)$ be as in your definition. Then $F(x)=\int_0^x f(t)\,dt$ exists and is $0$ for all $x$. Certainly $F(x)$ is differentiable for all $x$, but $F'(1)=0$ while $f(1)=1$. (You were asked to prove or disprove $A$ **and** $B$. You have disproved it if you show $B$ fails.)2012-04-05
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    Your function is certainly Riemann integrable (only countably many discontinuities). The problem with $F^\prime(1/2)$ is immediate. If $f(x)$ were in addition continuous though, the statement would hold. But isn't $f(1/2) = 1/2$?2012-04-05
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    A true theorem never fails.2012-04-06

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