8.3 A Non-integrable Function

Then , since if where and are odd and is even, then which is impossible since is odd and is even.

Proof: Since we can choose an odd integer such that
, i.e., . Since the interval has length
, it
contains at least two integers , say . If and are both
odd, then there is an even integer between them, and if and are both
even,
there is an odd integer between them, so in all cases we can find a set of
integers
one of which is even and the other is odd such that ,
i.e.,
. Then
and
are
two elements of one of which
is in , and the other of which is in .

I will find two partition-sample sequences and such that

and

It then follows that is not integrable. Let be the regular partition of into equal subintervals.

Let be a sample for such that each point in is in and let be a sample for such that each point in is in . (We can find such samples by lemma 8.35.) Then for all

and

So and . Both and are partition-sample sequences for , so it follows that is not integrable.

Our example of a non-integrable function is a slightly modified version of an example given by P. G. Lejeune Dirichlet (1805-1859) in 1837. Dirichlet's example was not presented as an example of a non-integrable function (since the definition of integrability in our sense had not yet been given), but rather as an example of how badly behaved a function can be. Before Dirichlet, functions that were this pathological had not been thought of as being functions. It was examples like this that motivated Riemann to define precisely what class of functions are well enough behaved so that we can prove things about them.