I'm looking for some continuous functions $\{f\_i(x,t)\}$, here $x=(x\_1,x\_2..., x\_n)$, such that:


* $f\_i(x,t):R^n\times [0, +\infty)\rightarrow R ~~\text{is continuous for each}~~i $
* $f\_i(x,0)=x\_i$
* $\Sigma\_1^n f\_i^2(x,t)\rightarrow \infty ~~\text{as}~~\|(x,t)\|^2=\Sigma\_1^n x\_i+t^2 \rightarrow \infty$


If such functions exist, could you please show me some examples, thanks.



Looking for some special functions
I assume you want squares on $x\_i$ in the expression for the norm: $\|(x,t)\|^2=x\_1^2+\dots+x\_n^2+t^2$.


Then I believe there are no such functions for topological reasons. Consider the restriction of $f=(f\_1,\dots,f\_n)$ to a half-sphere
$$X=\{(x,t);\,x\_1^2+\dots+x\_n^2+t^2=C,\,t\geq 0\}.$$
Suppose that $f\neq 0$ on $X$. Then, $f/|f|$ would be a continuous map from $X$ to the unit sphere $S^{n-1}$ in $\mathbb R^n$. Moreover, the restriction to the boundary $t=0$ is $f(x,0)=x/|x|$, that is, it is the identity under the natural identification $\partial X\approx S^{n-1}$. My understanding of algebraic topology is near zero, but I believe it follows from homotopy theory that no such map exists. So $f$ has a zero on $X$ for any $C>0$, which contradicts your last assumption.


