Jumps lead to the Gibbs phenomenon and lots of ripples that hurt global polynomial fits. Smoothing splines do quite a bit better in these cases. Effects for choosing the smoothing parameter It can seem in examples that smoothing splines have little advantage over simpler methods, like polynomial regression. For relatively smooth functions, polynomial regression has much smaller MSE than smoothing splines. The advantages only become visible when you have the right sort of test function, particularly one that has a jump.
Jumps lead to the Gibbs phenomenon and lots of ripples that hurt global polynomial fits.
Smoothing splines do quite a bit better in these cases. Picking the smoothing parameter from data Simulation lets you see how well a procedure can do, and how its MSE is determined by the smoothing parameter. What it does not show is how to choose this parameter from data, without the benefit of knowing the test function. You really only become challenged if you need to simulate such an estimator. You either have a very smooth fit or a very rough one, but you cannot mix both.