I'll throw in my funky function (which is probably similar to a well-known one, but I'm no mathematician so I know not what is well known):

f(x) = a random number uniformly selected from [-1, 1].

(So every real number maps to a distinct random number.)

It's nowhere continuous, nowhere differentiable, yet its integral over any interval is exactly zero.

Which is kind of weird if you think of integrals and derivatives as inverses, because as I understand it the derivative of a constant g(x) = c is g'(x) = 0, yet the integral of the above function is F(x) = 0 yet F'(x) = f(x) = the random nonsense monster. So the derivative of zero can be either zero or random noise? (And while f(x) is not differentiable, its integral surely is.)