real analysis - Continuous and differentiable functions

I know the definition of continuous and differentiable function from $\mathbb{R}$ to $\mathbb{R}$, but I am confused about how to apply that to a function from $\mathbb{R}^{m}$ to $\mathbb{R}$ like
$f(u) = u^T Q u$ where $u$ is a $m \times 1$ vector and $Q$ is a $m \times m$ matrix. Is function $f$ continuous and differentiable? Does the answer depend on the characteristics of matrix $Q$ (i.e., positive definite, positive semi-definite,...)?

Thanks,

Answer

A function $f:\mathbb R^n\to\mathbb R$ looks like $f(x_1,...,x_n)$. We say that $f$ is continuously differentiable if $$\frac{\partial f}{\partial x_i}(x_1,...,x_n)$$ exists and is continuous for all $i=1,...,n$. The example you gave is a quadratic form, meaning that if you do the matrix multiplication, you end up with $$f(x_1,...,x_n)=x^TQx=\sum_{i,j=1}^n q_{ij}x_i x_j.$$ Since the partial derivatives of $f$ are just the partial derivatives of each term in the expansion added together, it is not hard to see that all the partial derivatives exist and are continuous, regardless of the form of $Q$.