If the absolute value of a function is Riemann Integrable, then is the function itself integrable? I am trying to check the converses of a few theorems.
I know that that if $g$ is integrable then $|g|$ is integrable. However, if $|g|$ is Riemann Integrable, then is $g$ Rieman integrable?
I know that if $g$ is integrable then $g^2$ is integrable. However, is the converse true?
I have a hunch that they aren't true, but am failing to device the counter examples.

Let $f(x)=1$ when $x$ is rational, $-1$ when $x$ is irrational. Interval say $[0,1]$. 
