Functions That Cannot Be Integrated over a Closed Interval: The Dirichlet Function
Definition
The Dirichlet function is defined as follows: $f$
Explanation
The Dirichlet function is a classic example of a function that can’t be integrated using Riemann integration. Unless one advances in the study beyond analysis, it’s unlikely to even imagine such a peculiar example. The specific mention that it can’t be integrated using Riemann integration is because it may be integrable by methods other than Riemann integration.
Theorem
The Dirichlet function can’t be integrated over $[0,1]$.
Proof
Due to the density of real numbers, no matter how we partition $P$, the upper Riemann sum is $U ( f , P ) = 1$ and the lower Riemann sum is $L (f, P) = 0$. Therefore, it is not integrable.
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