How smooth are the singular values of a matrix F in terms of entries of F? I am hoping for Lipschitz continuity, but was not able to find it.
The singular values of $F$ are the (square roots of ) eigenvalues of $F F^t,$ and the regularity of the latter have been studied halftodeath. See either T. Kato (perturbation theory of linear operators, ch. 1) or Golubvan Loan (Matrix Computations  they almost certainly talk about singular values directly, without going through eigenvalues, but at worst talk about eigenvalues). 


Check these references 

