Continuity of the determinant function

[dextercioby]

This is something I seek a proof of.

Theorem: Let $\mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}$ be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous.

My attempt. Continuity must be judged in $\mathbb{R}$, so it should eventually go down to an epsilon-delta proof. I was first thinking to identify $\mbox{Mat}_{n\times n}(\mathbb{R}) = \mathbb{R}^{n^2}$. How do I go further?

 

[lavinia]

This is something I seek a proof of.

Theorem: Let $\mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}$ be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous.

My attempt. Continuity must be judged in $\mathbb{R}$, so it should eventually go down to an epsilon-delta proof. I was first thinking to identify $\mbox{Mat}_{n\times n}(\mathbb{R}) = \mathbb{R}^{n^2}$. How do I go further?

Note that the determinant is a polynomial in the $n^2$ coordinates of $R^{n^2}$. For instance for a $R^4$ it is $xw-yz$.