There was a great programme on Radio 4 yesterday about Newton and Leibnitz. I'm not sure why, but it inspired me to consider the curve that is constructed from y = x2 when x <= 0, and y = x when x > 0 . I want to reduce that to a single function.
It can obviously be done.
Consider that to fit a curve to any* three points, you can do it with a curve y = a + bx + cx2. Figuring out the three constants a, b and c is trivial, you just solve three simultaneous equations. Similarly, for four points, you fit y = a + bx + cx2 + dx3, and so on. To accurately fit the curve I described at the beginning, you need an infinite number of points, thus an infinite number of constants.
I'm sure that this can be done - obviously you can't actually calculate an infinite number of constants, but I'm sure that with a bit of integration it could be done. And it can be done for any such pair of functions which meet at a point. However, on further reflection I'm not entirely convinced that it can be done in the general case - you have that pesky discontinuous yes/no conditional in the middle: "is x <= n?".
* not strictly true - consider (0,0), (0,1), (1,0).