Suppose $f$ is a real function on $(0, 1]$ and $f \in \mathscr{R}$ on $[c, 1]$ for every $c > 0$. Define $$ \int_0^1 f(x) \ \mathrm{d} x = \lim_{c \to 0} \int_c^1 f(x) \ \mathrm{d} x $$ if this limit exists (and is finite). (a) If $f \in \mathscr{R}$ on $[0, 1]$, show that this definition of the integral agrees with the old one.
Question(1) I don’t understand the given definition. Like $f$ is defined on $(0,1]$, how can we talk about $\int_0^1 f(x) \ \mathrm{d} x$? Because by definition 6.1, $f$ needs to be defined on $[0,1]$. Do we have to assume $f \in \mathscr{R}$ on $[0,1]$?
Question(2) I don’t know precisely what to prove. Can someone please tell, in term of If...then... format, what to prove?