Classify regularity

The various classes of regularity are defined starting from the reiterated derived from the functions, and the possible continuity of these derivative.

If $I \, \!$ is a interval of $\ R \, \!$ , and $k \ geq 1$ an entirety one notes $\ mathcal {D} ^k (I, \ R) \, \!$ the whole of the functions of $I \, \!$ towards $\ R \, \!$ which is $k \, \!$ derivable times. One notes $\ mathcal {C} ^k (I, \ R) \, \!$ the subset of $\ mathcal {D} ^k (I, \ R) \, \!$ formed by the functions of which the $k \, \!$ -ième derived is continuous, and one notes $\ mathcal {C} ^0 (I, \ R) \, \!$ the whole of the continuous functions of $I \, \!$ towards $\ R \, \!$ . Finally one notes $\ mathcal {C} ^ {\ infty} (I, \ R) \, \!$ the whole of the indefinitely derivable functions of $I \, \!$ towards $\ R \, \!$ .

These units are algebras on $\ R \, \!$ and one following inclusions:

$\ mathcal {C} ^0 (I, \ R) \ supset \ mathcal {D} ^1 (I, \ R) \ supset \ mathcal {C} ^1 (I, \ R) \ supset \ mathcal {D} ^2 (I, \ R) \ supset \ mathcal {C} ^2 (I, \ R) \ supset \ cdot \ cdot \ cdot \ supset \ mathcal {D} ^k (I, \ R) \ supset \ mathcal {C} ^k (I, \ R) \ supset \ mathcal {D} ^ {k+1} (I, \R) \ supset \ mathcal {C} ^ {k+1} (I, \ R) \ supset \ cdot \ cdot \ cdot \ supset \ mathcal {C} ^ {\ infty} (I, \ R) \, \!$ .

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