From:Principles of math analysis by Rudin, Chapter 6 Problem 7
Suppose $f$ is a real function on $(0, 1]$ and $f \in \mathscr{R}$ on $[c,1]$ for every $c>0$. Define $\int_0^1 f(x)dx=\lim_{c\to 0} \int_c^1 f(x)dx$ if this limit exists (and is finite). If $f \in \mathscr{R}$ on $[0,1]$, show that this definition of the integral agrees with the old one.
Can I use the fundamental theorem of calculus to prove this problem? $$|\int_c^1 f(x)dx-\int_0^1 f(x)dx|=|\int_0^c f(x)dx|$$, and define $$F(c)=\int_0^c f(x)dx$$ is continuous so $$\lim_{c\to 0}|\int_0^c f(x)dx|=|\lim_{c\to 0}F(c)|=0$$
