I think precision would require you to state this in terms of a variable x and the function f(x).

gjm has read the note I linked; I suggest you do the same. That is what a link is for.

This is a pretty harsh requirement!

Not particularly. The speed of a car, the temperature of a room, the height of an aircraft: such things are all around you. Stating the property of the whole real line is an idealisation, but Theorem 1 of the note treats of finite intervals also, and there is a version of the theorems for time series.

I don't think you can say it's true for y=x.

In keeping with the terminology established at the note I linked, I take this to mean x=t. Yes, it is not true of x=t. This does not have an average over the whole real line.