Suppose I have a bounded region in the complex plane, M. I want to show that a point P is in the interior of this region.
At first I thought that if through P, for any arbitrary direction, I could find a continuous set of points within M whose tangent at P was parallel to that direction, then that would be sufficient to prove P is an interior point.
But this isn't true. Take for example two filled circles tangentially intersecting at a single point P. Through P I can move infinitesimally in any direction, but P is a boundary point.
But what if I could prove that the same property of being able to move infinitesimally in any direction and remain in the region applies to all points "near" P. Would that be sufficient to prove P is in the interior?
Is there any applicable theorem in topology here I can use?