In quantum mechanics and gravity theories one always meets the word "singularity" in connection with talks about black holes and the big bang, etc. Now in mathematics a singularity is well defined – mostly as a pole of $n$-th order that makes the function value approach infinity near the singular point...
But I cannot imagine that a singularity can actually exist in the real world. It is simply "un-physical" and any equations having physical meaningful singularities must be wrong or the singularity somehow excluded just like saying $v/c$ is always less than 1.
Hence, a black hole can not in it self "be" a singularity but might behave as such near that point, which, however, nothing ever reaches. Thus it can not be observed and is physically non-existent as a point in time and space but only an abstraction – similar to what I would be tempted to say about the photon.
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