Give an example of function which is bounded on [a,b] but not Riemann integrable on [a.b].

Question

Unable to find the function which is bounded but not Riemann integrable.

Answer 1

You need a highly discontinuous function. The classical example is $$ f(x)=\cases{0& if $x\in\Bbb Q$\\1& if $x\notin \Bbb Q$} $$

Answer 2

Take any nice function, like $f(x) = 1$ or $f(x) = x$ and have it take different values on the irrationals. This almost always works for this type of thing. This is (AFAIK) what bothers people about the Riemann integral.

$$f(x) = 1$$ if $x$ is rational and $$f(x) = 0$$ otherwise.