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**Mollier** Hi, the way I understand it, the first part of the fundamental theorem of calculus says that every function that is continuous on the closed interval $\displaystyle [a,b]$ has an antiderivative. If we denote the antiderivative by $\displaystyle F$ it looks like,

$\displaystyle F(x) = \int^x_a f(t)dt$.

Now, F is continuous on $\displaystyle [a,b]$ and differentiable on $\displaystyle (a,b)$, and $\displaystyle F'(x)=f(x)$ for all $\displaystyle x$ in $\displaystyle (a,b)$.

*Why is $\displaystyle F$ not differentiable on the closed interval $\displaystyle [a,b]$?*