This is the concept of external validation of internal processes, which is part of the problem with the inherent solipsism of the question.
There's no way to externally validate that the you inside is the same.
Just as if you were copied in the teleporter with one destroyed and the other created, your friends and family and videos of you would match the before teleporter and after teleporter versions, even though the old one was dead and the other hadn't existed.
You just kind of have to just go on belief that the you inside is continuous. There is no way to measure it to validate, as there's currently no agreed upon measurement of consciousness in neuroscience even.