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?