Let x be a differentiable real function.

I think precision would require you to state this in terms of a variable x and the function f(x). (EDIT: Sorry; please ignore this.)

If the averages of x and dx/dt over the whole real line exist,

This is a pretty harsh requirement! It will be true for constant functions, cyclic functions, symmetric functions, and maybe asymptotically-bounded functions. I don't think you can say it's true for y=x.