On continuity of self

Continuity is a tricky issue. To understand that the feeling of continuity - the feeling that I existed for so long- arises due to the desire to exist, as Buddha puts it due to bhava tanha (desire to become) is a revealing insight. Having said that, if one believes, by wrongly grasping Anatta as a principle, that there is no 'I'; only dhammas exist, is also equally wrong. It is actually vibava tanha, desire for non existence. But it is inherent that we tend to believe that the 'I' the enemy and tend to ignore it.
The result is that when faced with a tough situation, we tend to solve the situation with immediate instincts, which is the reaction, the only arsenal we have at hand. We don't want to bring our good nature, our strengths that we acquired when facing the issue. Inherently we miss the continuity of our personality !
To note that it's the 'I' feeling that brings in our good conditioning and our past strengths is a wonderful insight, that's the key.
The trick is that what inherently we should not hold on to as oneself is the accumulated past actions and their effects on the psyche. What one should hold on to oneself is the feeling of oneself. The latter is inherently just a feeling but it is a true feeling.
Ramana maharishi says that holding on to this feeling and seeking leads to the beyond. That is for one to see.