# Derivating $f(t)=\int_0^t x dx$ using measure theory

For the function $f(t)=\int_0^t x dx$, Riemann fundamental theorem of calculus says $$f'(t)=t$$

On Lebesgue side, I know a theorem says $$f'(t)=t \text{, on } \mathbb R \setminus E \text{ where } E \text{ has measure } 0.$$ I think this is weaker than Riemann results as I don't know which point are in $E$.
Is there a way, from Lebesgue point of view, to know the point in $E$?

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Yes. You know that $E$ is empty because the integrand is a continuous function on the closed and bounded set $[0,t]$ so it's Riemann integrable and its Riemann integral agrees with its Lebesgue integral.