1. Let $M=\{(x, y) \mid x=\cos \theta, y=\sin \theta, 0<\theta<\pi\}, N=\{(x, y) \mid$ $y=x+m\}$. If $M \cap N \neq \varnothing$, then the range of real number $m$ is
1. $-\frac{\sqrt{2}}{2}<m \leqslant 1$.

If $M \cap N \neq \varnothing$, then $\sin \theta=\cos \theta+m, m=\sqrt{2} \sin \left(\theta-\frac{\pi}{4}\right)$. And $\theta \in(0, \pi), \theta-\frac{\pi}{4} \in \left(-\frac{\pi}{4}, \frac{3}{4} \pi\right), m \in\left(-\frac{\sqrt{2}}{2}, 1\right]$, hence the range of $m$ is $-\frac{\sqrt{2}}{2}<m \leqslant 1$.