Properties of piece-wise fits and regression splines: (a) For the regression equation and show that the regression lines in the three bins are continuous (i.e., join at knots on the bin boundaries at...

Properties of piece-wise fits and regression splines:

(a) For the regression equation

and

show that the regression lines in the three bins are continuous (i.e., join at knots on the bin boundaries at X = k₁
and X = k₂).

(b) Similarly, show that the cubics in the regression equation

are continuous, that the cubics in the regression equation

are both continuous and have continuous slopes at the knots, and that the cubics in the regression-spline equation

join at the knots, have continuous slopes, and have continuous curvature.