This is a problem No.43 of the chapter 5 exercises of Pugh's real mathematical analysis.
Suppose that T : R$^n$ $\to$ R$^m$ has rank k.
(a) Show there exist a $\delta$ $>$ $0$ such that if S : R$^n$ $\to$ R$^m$ and $||$S$-$T$||$$<$$\delta$ then S has rank $\geq $ k.
(For the notations, T and S are linear operators, and $||$A$||$ where A is a matrix is a supremum metric.
For example, $|$Ax$|$ $\le$ $||$A$||$$|$x$|$
I have been thinking about it for hours, but I can't make it. Help me.)