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What are piecewise-$C^k$ functions? How can I know if a function is piecewise-$C^k$ or not?

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A piecewise-$C^{k}$ is a function whose domain can be broken into intervals, upon which the restriction to each individual interval is $C^k$ (but that might fail at endpoints). – Isaac Solomon Mar 7 '13 at 18:06
@IsaacSolomon What's a $C^k$ interval? – saadtaame Mar 7 '13 at 18:11
I think it's not $C^k$ interval but in that interval the function is $k$ times continuously differentiable. – user45099 Mar 7 '13 at 18:25

Any function with k continuous derivatives is called a $C^k$ function. And look here as to what piecewise means-

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