Let $f:X\rightarrow Y$ be a diffeomorphism between connected oriented manifolds. $f$ is orientation-preserving at $p\in X$ if the induced map $df_{p}:T_{p}X\rightarrow T_{f(p)}Y$ is orientation-preserving; similarly $f$ is orientation-reversing at $p$ if the derivative is orientation-reversing. Why must $f$ be either orientation-preserving everywhere or orientation-reversing everywhere?
I think it is true that the sets of points where $p$ is orientation-preserving and orientation-reversing are both open (which implies the result), but I can't prove this.