Certain properties of analysis are stated with functions checking of the assumptions such as continuous per pieces , of class $\backslash \; mathcal\; C^k$ per pieces , etc

Regularity per pieces on a segment

A function F is continuous per pieces on the segment when there exists a subdivision F with each open interval ] a_i, a_i+1 admits a prolongation [[function continuous|continuous] with the interval closed .

Concretely the function F is continuous on] a_i, a_i+1 admits one [[limit] on the right and on the left in each a_i, which can be distinct and distinct from the value from F at the point a_i itself.

One defines in the same way the functions of class $\backslash \; mathcal\; C^k$ per pieces , linear per pieces, etc

It will be noted that a function of class $\backslash \; mathcal\; C^1$ per pieces, for example, is not necessarily continuous in a_i, but that she admits limits and Dérivée S on the right and on the left in a_i.

Regularity per pieces on an interval

A function is continuous (or other properties) per pieces on the interval I when it is continuous (or other) per pieces on any segment of I .

Fields where one uses the regularity per pieces

• Certain simplified theories of the integration

• the Fourier series