Let $\delta > 0$ be chosen so that $$\|x - a\| < \delta \Rightarrow \left\vert\frac{\partial^2 f}{\partial x_i \partial x_j}(x) + \frac{\partial ^2 f}{\partial x_i x_j}(a) \right\vert < \epsilon.$$
Then consider $$\vert v^T (D^2 f(x) - D^2f(a)) v\vert$$ where $D^2$ is the Hessian matrix and $v$ is some vector. Why is above less than $k \epsilon \| v\| ^2$ for some $k$?
I know that $$\vert v^T C v\vert < \sqrt{\sum c_{ij}^2} \|v \| ^2$$ where $C$ is some matrix and $c_{ij}$ are its components. But the 'absolute value' signs disappear and so I can't use my continuity condition, which requires an absolute value sign?