Suppose $f:X\to Y$ is a diffeomorphism of connected oriented manifolds with boundary. Show that if $df_x: T_x(X)\to T_{f(x)}(Y)$ preserves orientation at one point, then $f$ preserves orientation globally.
What I can see is that since $f$ is a diffeomorphism, each $df_x$ is an isomorphism. But the concept of orientation preserving linear map remains not so clear to me (see this question). Is the fact that $df_x$ preserves orientation amounts to saying that $\det df_x > 0$? After that, am I supposed to use some kind of continuity of $x\mapsto df_x$?