  An $(r,n)$-velocity is an $r$-jet with source at $0 \in \R^n$, and target in
a manifold $Y$. An $(r,n)$-velocity is said to be regular, if it has a
representative which is an immersion at $0 \in \R^{n}$. The manifold
$T^{r}_{n}Y$ of $(r,n)$-velocities as well as its open, $L^{r}_{n}$-invariant,
dense submanifold $\Imm T^{r}_{n}Y$ of regular $(r,n)$-velocities, are endowed
with a natural action of the differential group $L^{r}_{n}$ of invertible
$r$-jets with source and target $0 \in \R^{n}$. In this paper, we describe all
continuous, $L^{r}_{n}$-invariant, real-valued functions on $T^{r}_{n}Y$ and
$\Imm T^{r}_{n}Y$. We find local bases of $L^{r}_{n}$-invariants on $\Imm
T^{r}_{n}Y$ in an explicit, recurrent form. To this purpose, higher order
Grassmann bundles are considered as the corresponding quotients $P^{r}_{n}Y =
\Imm T^{r}_{n}Y/L^{r}_{n}$, and their basic properties are studied. We show
that nontrivial $L^{r}_{n}$-invariants on $\Imm T^{r}_{n}Y$ cannot be
continuously extended onto $T^{r}_{n}Y$.
