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If we have a regression spline of d degree of freedom then why it is neccessary to have its first d-1 derivatives be continous ?

For example if we have a cubic spline then why it is neccessary to have its first two derivatives (i.e. First derivatives and Second derivatives) to be continuous ?

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    $\begingroup$ My guess is that's part of the definition of a spline. If you want discontinuous predictions go for it (think: a small change in the inputs leads to drastic changes in predictions - do you want this) $\endgroup$
    – probabilityislogic
    Commented Apr 10, 2016 at 14:17

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You can fit polynomials with whatever order of continuity of derivatives you like (including not having any of them continuous), but if it doesn't have the derivatives being continuous to the required order, it's not strictly a spline.

This is because continuity of derivatives to a particular order is part of the definition of a spline.

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