Continuity of a $dx_1 \wedge \dotsb dx_n(Y_1,\dotsb, Y_n)$, where $(Y_1, \dotsb, Y_n)$ is a continuous local frame

Question

Continuity of a $dx_1 \wedge \dotsb dx_n(Y_1,\dotsb, Y_n)$, where $(Y_1, \dotsb, Y_n)$ is a continuous local frame on an open subset $U$ of a smooth manifold $M$. $dx_1 \wedge \dotsb dx_n(Y_1,\dotsb, Y_n)$ is defined pointwsiely by $dx_1 \wedge \dotsb dx_n(Y_1,\dotsb, Y_n)|_p = dx_1 \wedge \dotsb dx_n|_p(Y_1|_p,\dotsb, Y_n|_p)$. Why should such a function be continuous?

Answer 1

If $(Y_1,\dots,Y_n)$ is a continuous local frame on $U$, we can write each $Y_i$ in terms of the frame $\partial_i = \frac{\partial}{\partial x^i}$ as $Y_i = T_i^j \partial_j$ where $T_i^j$ are continuous functions on $U$. That is, each $Y_i$ is a continuous section of $TM$ over $U$ and by recalling the definition of the topology on $TM$, we see that this implies that the functions $T_i^j$ are continuous. Now,

$$ (dx^1 \wedge \dots dx^n)(Y_1, \dots, Y_n) = \det(T_i^j) $$

is also continuous.

More generally, you can prove the same way that if $\omega$ is a continuous differential $k$-form (that is, $\omega$ is a continuous section of $\Lambda^k(T^{*}M)$) and $Y_1,\dots,Y_k$ are continuous vector fields then $\omega(Y_1,\dots,Y_k)$ is a continuous function.