Is this transformation $\mathbb{R}^{n \times n} \ni A \to \det A \in \mathbb{R}$ continuous?

Question

Let $\mathbb{R}^{n \times n}$ be provided with the following norm: $$\|A\|_{\mathrm{HS}}=\left(\sum_{i=1}^n\sum_{j=1}^n|a_{i,j}|^2\right)^{1/2},\ A=(a_{i,j})_{i,j=1,\dots,n}\in\mathbb R^{n×n}$$ How can one prove that the transformation $\mathbb{R}^{n \times n} \ni A \to \det A \in \mathbb{R}$ is continuous?

I can identify ($\mathbb{R}^{n \times n}, \left\lVert \cdot \right\rVert_{HS}$) with ($\mathbb{R}^{n^2}, \left\lVert \cdot \right\rVert_{2}$) and when I look at the exponent, which is $1/2$, I could take the square root of the sum, but I really don't know how to deal with this...

Answer 1

Using Laplace expansion you can see that the determinant is just an homogeneous polynomial, thus it must be a continuous function.