On Thu, 3 Jul 2008, Matt Barber wrote:
I'm with you on the general piecewise Lagrange not being C1, but I don't think it follows that all other orders are discontinuous -- can't they alternate?
No. Any derivative inherits all discontinuities from the original function. It's a basic fact of life.
Therefore, since the terms match at the points on adjacent pieces, the 2nd derivative is continuous even though the first isn't.
Ok, well, there's usually a distinction made between a continuous function and one with holes. Even though you can use a limit operator to remove the holes, the result of that is just not the same function... unless you see that through limit-operator-coloured glasses.
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