\begin{cases}\cos\left(\frac{\pi}{x}\right) &,\quad \text{if } x \text{ is rational}\\0 &,\quad \text{if } x \text{ is irrational}\end{cases}
Over the interval $[0,1]$
My approach
Solve for upper and lower integral. But was unable to somehow account for the 3 different types of intervals possible(both rational and irrational $x$ , only rational $x$ and only irrational $x$)
I was unable to find a partition of $f(x)$ for which $U(p,f) - L(p, f) < ε$ is NOT satisfied. (I tried for $[0.4, 0.5]$)