Are Individual Functions Riemann Integrable if Their Sum is?

[losin]

suppose f and g are bounded functions on [a,b] such that f+g is in R[a,b]

Then, does it follow that f and g are also in R[a,b]? i wanto to prove whether it is or not

 

[mathman]

Not so. Let f be a function which is NOT Riemann integrable and let g=1-f.

For example f(x)=1, when x is irrational and f(x)= 0, when x is rational.