Question

How do I find one-sided limits of piecewise functions?

Answer 1

If x = c is the junction for the two pieces `y = f(x), x in (a, c)` and `y = g(x), x in (c, b)`, the one-sided limits at the junction x = c are `h to 0` of `f(c-h) and g(c+h)`.

Explanation:

The piecewise functions are jointly continuous at x = c, if the one-sided limits are the same.

Further if the one-sided limits for f' ad g' are the same, they are differentiable at x = c.

For piece-wise cubic-spline fitting, this matching is extended up to f'' and g''.