Thomae Function
in lowest terms and at irrationals; lower sums vanish and upper sums can be made arbitrarily small, so is integrable on despite being discontinuous at infinitely many points.
Consider the function defined by
Since is dense in , for any partition of , hence . We claim that is integrable, then such that .
Pick such that . Set
Define such that
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each is some subinterval of
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this subinterval has length [we wish to give little weight to the bad points]
Then,
Contrast with the Dirichlet function
The integrability proof: given , pick with and set
for some finite . Choose a partition in which each lies in a subinterval of length , giving the bad points little weight. Then
while always .
Compared with the Dirichlet function, the values of the Thomae function at its bad points decay to : both functions have infinitely many discontinuities, but their integrability properties are fundamentally different.
Related
Stated in
- Example 4.16 (Thomae Function)§4.2 Integrability Criteria