Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$ let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for 
$$
\|
\sigma_i(A)-\sigma_i(B)
\|
$$
as a function of $\|A-B\|$ (for some matrix norm $\|\cdot\|$)?