Suppose a nonlinear infinitely continous differentiable function $f:\mathbb{D}\mapsto \mathbb{R^+}$, where $\mathbb{D}\subset\left\{X|\text{rank}{X}=2,X\in\mathbb{R}^{3\times 3}\right\}$ is a continous, open, convex set,

How to define a matrix norm $\left\|\cdot\right\|_p$ in $\mathbb{D}$, such that $\forall x_1, x_2\in \mathbb{D}$: $$\lim\limits_{\left\|x_1-x_2\right\|_p\to 0}|f(x_1)-f(x_2)|=0,$$ $$\lim\limits_{\left\|x_1-x_2\right\|_p\to 0}\left\|x_1-x_2\right\|_2=0$$ $$\quad\lim\limits_{\left\|x_1-x_2\right\|_p\to 0}\left\|x_1-x_2\right\|_{\infty}=0,$$ $$\lim\limits_{\left\|x_1-x_2\right\|_p\to 0}\left\|x_1-x_2\right\|_1=0,$$

and the $9\times 1$ vectors $y_1 = x_1(:),y_2=x_2(:)$(use Matlab/Octave operator here) $$\lim\limits_{\left\|x_1-x_2\right\|_p\to 0}\left\|y_1-y_2\right\|_2=0,$$

Must the calculation of such matrix norm be `NP`

-hard?

continuous set? $\endgroup$`continuous`

, I mean`connected`

$\endgroup$