How do I find one-sided limits of piecewise functions?
If x = c is the junction for the two pieces
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''.