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Let $W$ be a finite reflection group. A $W$-invariant function ofclass~$C^{\infty}$ may be expressed as a functions of class $C^{\infty}$ of thebasic invariants. In finite class of differentiability, the situation is notthis simple. Let~$h$ be the greatest Coxeter number of the irreduciblecomponents of $W$ and $P$ be~the Chevalley mapping, if $f$ is an invariantfunction of class $C^{hr}$, and $F$ is the function of invariants associated by$f=F\circ P$, then $F$ is of class $C^r$. However if~$F$ is of class $C^r$, ingeneral $f=F\circ P$ is of class $C^r$ and not of class $C^{hr}$. Here wedetermine the space of $W$-invariant functions that may be written as functionsof class $C^r$ of the polynomial invariants and the subspace of functions $F$of class $C^r$ of the invariants such that the invariant function $f=F\circ P$is of class $C^{hr}$.

Finite Reflection Groups: Invariant functions and functions of the Invariants in finite class of differentiability