It is hard to find any reference which contains a proof of the following statement: slope stability is an open condition in a flat family. There is one I have found, '65 paper of Narasimhan and Seshadri. Is this the unique reference which proves the statement? There are some references which prove the openness of Gieseker stability, but what I want is about slope stability. I will be appreciate if one introduces a reference other than Narasimhan and Seshadri...
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In the second edition of "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn, the arguments in Proposition 2.3.1 which imply (for instance) that Gieseker-semistability is an open condition in families also imply the same for $\mu$-stability (this is pointed out just after Definition 3.A.1 on p. 82).
