I need to find a Riemann integrable function, which is not monotonic on a closed interval. But I couldn't find one. I checked some continuous and discontinuous functions but still couldn't find a function satisfying the above things.
Can anyone give an example for Riemann integrable function, which is not monotonic on a closed interval?
Thanks in advance!
Assuming you mean a function that is not monotonic anywhere, but still integrable, there is the Weierstrass function, which is continuous everywhere (and thus integrable everywhere), but is nowhere monotonic.
My favorite nowhere-differentiable continuous $f:[0,1]\to [0,1].$
Take the uniform limit $g$ of a sequence $(g_n:[0,1]\to [0,1]^2)_{n\in\Bbb N}$ of continuous functions,
where, for each $n$ and for any $a,b\in\Bbb N_0$ with $a<2^n$ and $b<2^n,$
there exists $t\in \Bbb N_0$ with $t<4^n,$ such that for all $n'\ge n,\;$ $\;g_{n'}$ maps $[t4^{-n},(t+1)4^{-n}]$ into $[(a2^{-n},(a+1)2^{-n}]\times [b2^{-n},(b+1)2^{-n}].$
Then $g:[0,1]\to [0,1]^2$ is the Peano-Hilbert space-filling curve.
For $g(t)=(x,y)$, let $f(t)=x.$ Then $f:[0,1]\to [0,1]$ is a continuous surjection. Now $f$ is not constant on any interval $J \subset [0,1]$ of non-zero length but for any $t\in J,$ the set $\{t'\in J: f(t')=f(t)\}$ is uncountable, so $f$ cannot be monotonic on $J.$
