I'd like a function to animate an object moving from point A to point B over time, such that it reaches B at some fixed time, but its position at any time is randomly perturbed in a continuous fashion, but never goes backwards. The objects move along straight lines, so I only need one dimension.

Mathematically, that means I'm looking for some continuous f(x), x ? [0,1], such that:

• f(0) = 0
• f(1) = 1
• x < y ? f(x) = f(y)
• At "most" points f(x + d) - f(x) bears no obvious relation to d. (The function is not uniformly increasing or otherwise predictable; I think that's also equivalent to saying no degree of derivative is a constant.)

Ideally, I would actually like some way to have a family of these functions, providing some seed state. I'd need at least 4 bits of seed (16 possible functions), for my current use, but since that's not much feel free to provide even more.

To avoid various issues with accumulation errors, I'd prefer the function not require any kind of internal state. That is, I want it to be a real function, not a programming "function".