Using fundamental theorem of calculus and differnetiability Say we have know that $$f(x)=\begin{cases} 1/n &amp;\text{if  $x=\frac{1}{n}$} \\ 0 &amp;\text{else} \\ \end{cases}$$ is $\in R[0,1]$ with a value of $0$.

and say we define $g(x)=xf(x)$, and $G(x)=\int_{0}^{x}g$

Then how could I prove that G is differentiable on $[0,1]$ but that $G'(c) \neq g(c)$ whenever g is not continuous at c, moreover if $F(x)= \int_{0}^{x}f$ then how could I prove it is differentiable on $[0,1]$ and that there exists $a,b$ in the interval with f being discontinuous at a and b and $F'(c)=f(c)$ but $F'(a) \neq f(a)$

My thoughts;

I am pretty confused. I know that g is R-int as it is the product of two R-int functions. I also know that f has a value of zero, so I think g will evaluate to zero. So G would be zero everywhere and hence differentiable, but I am not sure,
I cannot use results of Lebesque , ie about countable sets etc 

Does anyone at all have any advice please? I really dont know how to proceed and need help
I am still confused. Can anyone please help?
You're right.  $G(x) = \int_0^x g(y)\,dy = 0$ for all $x$.  Thus $G$ is differentiable but $G'(x) \ne g(x)$ for some $x$.  That is because $g(x)\ne 0$ only for $x=1/n$ with integral $n$, and that's just a countable set, and the integral of a function which is zero outside a countable set is zero.

I don't get what you mean by "with a value of $1$" at the end of your description of $f$.

Do you need a proof that a Riemann integrable function differing from $0$ on a countable set has Riemann integral zero?
You're right.  $G(x) = \int_0^x g(y)\,dy = 0$ for all $x$.  Thus $G$ is differentiable but $G'(x) \ne g(x)$ for some $x$.  That is because $g(x)\ne 0$ only for $x=1/n$ with integral $n$, and that's just a countable set, and the integral of a function which is zero outside a countable set is zero.

I don't get what you mean by "with a value of $1$" at the end of your description of $f$.

Do you need a proof that a Riemann integrable function differing from $0$ on a countable set has Riemann integral zero?