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Suppose $f$ is a real function on $(0, 1]$ and $f in mathscr{R}$ on $[c,1]$ for every $c>0$. Define $int_0^1 f(x)dx=lim_{cto 0} int_c^1 f(x)dx$ 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.

(b) Construct a function $f$ such that the above limit exists, although it fails to exist with $|f|$ in place of $f$.

This is Problem 7 of Chapter 6 in Principles of Mathematical Analysis by Rudin. For (a), I can prove the equation is correct but I am not sure what does 'definition agrees' mean? For (b), I have no idea.

Thank you in advance.

$b$) Using the well known integral
$$
int_{1}^infty frac{sin x}{x}mathrm dx
$$
which converges conditionally, we reflect reflect everything to near $0$ by sending
$$
xto 1/x
$$
and find
$$
int_{1}^inftyfrac{sin x}{x}mathrm dx=
int_0^1frac{sin(1/y)}{y}mathrm dy
$$

For part $a$, you must prove using the definition of the Riemann integral that for a function which is integrable on $[0,1]$, the number
$$
int_0^1 f(x)mathrm dx
$$
is the same as the number
$$
lim_{cto 0^+}int_c^1f(x)mathrm dx
$$