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Is this Dirichlet type function Riemann Integrable?

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\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

  1. 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 $x$)

  2. 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 $[0.4, 0.5]$)


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