Funció lineal definida a trossos

Una funció (en blau) i una aproximació lineal a trossos d'aquesta funció (en vermell).
Una funció lineal a trossos en dues dimensions (dalt) i els polítops convexes sobre els quals aquesta funció és lineal (a baix).

En les matemàtiques, una funció lineal definida a trossos

,

on V és un espai vectorial i és un subconjunt d'un espai vectorial, és una funció on és possible trobar una descomposició de en un conjunt finit de polítops convexos de manera que f sigui igual a una funció lineal en cadascun d'aquests polítops.

Un cas particular important és quan f és una funció real en un interval . En aquest cas f és lineal a trossos si i només si l'interval es pot particionar en un conjunt finit de subintervals de manera que sobre cadascun dels intervals I, f és igual a una funció lineal

f(x) = aIx + bI.

La funció valor absolut és un exemple de funció lineal a trossos. Les funcions part entera, ona dent de serra i ona quadrada són altres exemples d'aquest tipus de funcions.

Entre les subclasses de les funcions lineals a trossos cal esmentar les funcions contínues lineals a trossos i les funcions convexes lineals a trossos.

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