TITLE: When we say that "a function rarely is Riemann integrable", what exactly are we talking about?
QUESTION [8 upvotes]: I have read some authors saying that the class of all Riemann integrable functions is "quite small". In what sense? How can we prove this statement?

REPLY [5 votes]: Here're two ways to state it.
Claim 1: Let $B$ be the Banach space of all bounded functions on the unit interval under pointwise addition, scalar multiplication and sup norm: $\|f\| = \sup \{|f(x)| : x \in [0, 1]\}$. Then, the class of Riemann integrable (or even Lebesgue integrable) functions in $B$ is of first category (in fact, it is even closed nowhere dense).
Claim 2: Let $L$ be the Banach space of all bounded Lebesgue integrable functions on the unit interval under pointwise addition, scalar multiplication and sup norm: $\|f\| = \sup \{|f(x)| : x \in [0, 1]\}$. Then, the class of Riemann integrable functions in $B$ is of first category (in fact, even closed nowhere dense).
The choice of the norm in Claim 2 is flexible. You can use an $L^1$-norm and the statement remains true.