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In all calculus textbooks, after the part about successive derivatives, the $C^k$ class of functions is defined. The definition says :

A function is of class $C^k$ if it is differentiable $k$ times and the $k$-th derivative is continuous.

Wouldn't be more natural to define them to be the class of functions that are differentiable $k$ times? Why is the continuity of the $k$th derivative is so important so as to justify a specific definition?

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    Interesting question. Maybe an explanation is in the fundamental theorem of analysis: if $f$ is a $C^1$ function in the usual sense, then $f(b)-f(a)=\int_a^bf'(t)dt$. (it can be true for other classes of functions, but at least we are sure that the derivative is integrable).2012-07-16

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