I have a function, $f(y)$, which is continuous and bounded. I have the integral $$\int^{r}_{-r} f(y)\cdot sgn(y) dy$$
Which has been rewritten in my notes as
$$\int^{r}_{0} f(y)\cdot(1) dy + \int^{0}_{-r} f(y)\cdot(-1) dy$$
Is this a general rule for integrals over a symmetric range, involving the sign function? I imagine Reimann Integrability is also involved but I'm not entirely sure how this result is obvious. All help appreciated
This is a general rule for the Riemann integral breaking up an interval $[a,c]$ into $[a,b]$ and $[b,c]$ (with $a \leq b \leq c$).
One has $\int_a^c f(x)\,dx = \int_a^bf(x)\,dx + \int_b^c f(x)\,dx$, where the left side exists as a Riemann integral if and only if both the integrals on the right side exist as Riemann integrals. You can prove that if one of the integrals doesn't exist then one on the other side doesn't exist by definition using partitions with upper and lower sums.