## Not all functions are (Riemann) integrable!

1. After thinking for a moment, I should probably mention that not every function is (Riemann) integrable. Conversely, just because we throw something into an integral expression $\int\dots\,\mathrm{d}x$ doesn’t make it a function.

2. Consider the function
(1)$\displaystyle\chi_{\mathbb{Q}}(x) = \begin{cases}1 & x\mbox{ is rational}\\ 0 & \mbox{otherwise}\end{cases}$
What is its integral on the domain $[0,1]$? Well, using the Riemann sum, this depends on the partition. We can pick a partition whose end points are all rational; conversely we can pick one whose endpoints (except at 1 and 0) are irrational. This gives us radically different Riemann sums.

Consequently, we have to admit with a heavy sigh that it is impossible to integrate Equation (1) effectively.

3. On the other hand, there are times when we integrate stuff which aren’t functions. We won’t worry about what they are, or where they live, right now. Instead we will gleefully pretend everything is a function.

But an example of such an exotic creature. Well, the Dirac Delta “function”