"Pointwise squaring on an interval x" is just a weird way of describing the usual function f(x) = x^2 with domain restricted to an interval. It's pointwise because that's how functions f : R -> R are defined: given a point, or value, of the domain, give me a new point in the codomain.
If you think of `x` as a whole interval unto itself, and not just a single point, then I think the options become more interesting. The most natural product on two sets is indeed the cross product; but for intervals, I can imagine defining a common parameterization over both intervals and then multiplying pointwise up to that parameterization.
If you think of `x` as a whole interval unto itself, and not just a single point, then I think the options become more interesting. The most natural product on two sets is indeed the cross product; but for intervals, I can imagine defining a common parameterization over both intervals and then multiplying pointwise up to that parameterization.