Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and 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 $\|\cdot\|$)?