Definition of $C^r$ tensor fields in "Analysis on manifolds" by Munkres The text defines a $k-$tensor field as a function assigning, to each point $x$ in an open subset $A$ of $\mathbb{R}^n$, a $k-$tensor defined on the tangent space $T_x(\mathbb{R}^n)$, which can be written in the form $$\omega(x)((x;\mathbf{v_1}),...,(x;\mathbf{v_k}))$$
Then it requires this function to be continuous as a function of $(x,v_1,...,v_k)$ and, if it is of class $C^r$, it says that $\omega$ is a tensor field of class $C^r$.
My problem is with this last statement.
For each $x \in A$, $\omega(x)$ is a tensor. I don't really know what it means for a tensor to be continuous or differentiable.
Does it mean that $\omega$ should be continuous for $x$, continuous for $v_1$, etc. separately?
 A: A manifold in the sense of Munkres is a subspace $M$ of some $\mathbb R^n$ having suitable properties (see the definition in §23). in §29 Munkres also introduces the tangent space $T_x \mathbb R^n$ at $x \in \mathbb  R^n$ as the set $\{x\} \times \mathbb R^n$ which gets the structure of a vector space via the canonical bijection $\phi_x : \mathbb R^n \to T_x \mathbb R^n, \phi_x(v) = (x,v)$.
A $k$-tensor field on an open subset $A \subset \mathbb{R}^n$ (Munkres restricts to this case and does not define tensor fields on arbitrary manifolds $M \subset \mathbb R^n$) is defined as a function $\omega$ assigning to each point $x \in A$ a $k$-tensor $\omega(x)$ defined on the tangent space $T_x(\mathbb{R}^n)$. Thus (as he explicitly says) $\omega(x)$ is a function $(T_x \mathbb R^n)^k \to \mathbb R$. We therefore get a function
$$\bar \omega : A \times  (\mathbb R^n)^k \to \mathbb R, \bar \omega (x,v_1,\ldots, v_k) =  \omega(x)(\phi_x(v_1),\ldots, \phi_x(v_k)) .$$
But of course $A \times  (\mathbb R^n)^k$ is an open subset of $\mathbb R^{(k+1)n}$, thus it makes sense to call $\bar \omega$ continuous or $C^r$.
