TITLE: $\int_a^af(x) \, dx$ always $0$?
QUESTION [7 upvotes]: I was studying integrals and just out of curiosity,

Does there exist any 'continuous' functions such that $\int_a^af(x) \, dx$ ($a$ is any number) equals a value other than $0$?

Since continuous functions are Riemann integrable, so I think it should be $0$. Is this correct?
Also, with out the condition 'continuous', does there exist any function such that $\int_a^af(x) \, dx$ isn't $0$?
EDIT
I'm looking for any function that $\int_a ^a\ f(x) \neq 0$. Can anyone find me one?

REPLY [3 votes]: No, for $f(x)$ any continuous function, $\int_a^a f(x)dx= 0$.  As for non-continuous functions, my first thought was a "Dirac delta function" for which $\int_C \delta(x)dx= 1$ for any set, C, containing 0.  However, a "delta function" is not a true function- it is a "generalized function" or "distribution".