Smoothness of a discrete time signal

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Is it possible to find the smoothness ($C^1$, $C^2$,... continuity) of a time sampled signal?

For a continuous function continuity can be found by the number of existing derivatives. When does the derivative for a discrete time signal exist?

Deriving a discrete signal using e.g. finte differences is always possible I guess and therefore I was wondering how to find the continuity of a discrete signal?

In Matlab I thought of using dx = gradient(x) multiple times until I get "large peaks" above a predefined threshold, which would make it possible to define some sort of "pseudo smoothness" for a discrete signal.

Is there a better approach?