# Thread: need a counterexample for Riemann Integrable functions problem

1. ## need a counterexample for Riemann Integrable functions problem

I need to show that the following statement is false using a counterexample, problem is I cant come up with one!

Let f,h be Riemann integrable on the interval [a,b].
Let f(x)<=g(x)<=h(x) for all x in [a,b].
Then g is Riemann integrable.

any suggestions?

2. Originally Posted by dannyboycurtis
I need to show that the following statement is false using a counterexample, problem is I cant come up with one!

Let f,h be Riemann integrable on the interval [a,b].
Let f(x)<=g(x)<=h(x) for all x in [a,b].
Then g is Riemann integrable.

any suggestions?
on the interval $I=[0,1]$ define $f(x)=0, \ h(x) = 1, \ \forall x \in I.$ now define $g: I \longrightarrow \mathbb{R}$ by $g(x)= 1, \ \forall x \in I \cap \mathbb{Q},$ and $g(x)=0$ otherwise.