Is the nonsingular matrices open?
How can I show that every $m \times n$ matrix is in the image of the derivative of an $m \times n$ matrix (how to differentiate it?)
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Your first question is answered here (for the future, note that it is generally better to have one question per post, to keep things orderly). – Zev Chonoles Jul 07 '13 at 18:47
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This is true more generally in a unital Banach algebra, where we don't have the determinant to do that. The key, which works in any case, is that $I+A$ is invertible by the Neumann series whenever $|A|<1$ for a submultiplicative norm. It follows that if $A$ is invertible, then so is $A+B$ whenever $|B|<|A^{-1}|^{-1}$. – Julien Jul 07 '13 at 19:40