From the following theorem: Let $f$ be a continuous function which is positive on $[a,b]$. Then the corresponding area function has derivative i.e. $A'(x)=f(x)$ for all $x\in [a,b]$. The area function is antiderivative of $f(x)$. Then I have shown that if $f$ is continuous function which is positive on $[a,b]$. Then the area under the graph of $f$ equals $\int_a^b f(x) dx.$
Can I conclude from the above mentioned theorems that a continuous function has antiderivative?
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$\begingroup$The fundamental theorem of calculus says that for a continuous $f$ on $[a,b]$, the function $F(x)=\displaystyle\int_{a}^{x}f(t)dt$, $x\in[a,b]$ is an antiderivative of $f$.
$\endgroup$ 2 $\begingroup$Yes but not always you can find an expression by elementary functions.
E.G. $f(x)= e^{-x^2}$
$\endgroup$ $\begingroup$Basically, every continuous function is integrable and so are many non-continuous functions as well, according to the Reimann integral of infinite sum.
So, to put it simply, all the differentiable functions are subset of continuous functions, which themselves are subsets of integrable functions.
You can find more information about it here
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