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).
(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 
$$
x\to 1/x
$$
and find 
$$
\int_{1}^\infty\frac{\sin x}{x}\mathrm dx=
\int_0^1\frac{\sin(1/y)}{y}\mathrm dy
$$
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 
$$
\lim_{c\to 0^+}\int_c^1f(x)\mathrm dx
$$
