Show that there is a line segment with center of the point with assumed conditions

Question

Let $(x_0,y_0)$ be a regular point in phase portrait.Show that there is a line segment with center of this point such that all trajectories that cut the line segment have same direction from one of the sides of the line to an other and also there is not a trajectory that tangent to line segment.

I do not have a good idea to solve this.

Thank you.

Answer 1

Although you say nothing about what are the assumptions (space, vector field, anything else), continuity is all you need: near a regular point all points are regular and the vector field will maintain the sign with respect to any initial hyperplane.