Discontinuous Function and RS-Integral I saw the following exercise in Bartle’s Elements of Real Analysis:
If f is a function from the interval [a,b] to the reals such that f is discontinuous at some point of the interval, then there exists a monotone increasing function g such that f isn’t g-integrable. 
I understand that the g must be continuous at the point of discontinuity of the f for the f to be g-integrable. However, I don’t know how the proof should follow. 

Hints: for convenience, take $a=0$ and $b=1$ and suppose $f$ has a discontinuity at $x=1/2.$ 
$\text{Set}\  g(x)=\left\{  \begin{array}{lr} 0 &  x<1/2\\ \frac{1}{2} &  x=1/2\\ 1 & x>1/2 \end{array} \right\},$  note that $\text{osc}\ f=\epsilon>0$ at $x=1/2$ 
and prove that any interval containing $1/2$ satisfies $U(f,g)-L(f,g)>\epsilon.$
