I am trying to understand the proof of [Theorem 3.9, p.9](https://arxiv.org/pdf/math/0610266v1.pdf).

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We consider NLS $$i\partial_t u + \Delta + |u|^{4/(d-2)}u=0, u(x,0)=u_0 \in H^1(\mathbb R^d)$$
where $u:\mathbb R^{d+1} \to \mathbb C, u_0:\mathbb R^d \to \mathbb C$
 
Assume that    $\int_{\mathbb R^d} |\nabla u_0(x)|^2 dx \leq (1-\delta) \|\nabla W \|^2_{L^2}$  and $\int_{\mathbb R^d} |\nabla u_0(x)|^2 - |u_0(x)|^{2d/d-2} \geq \delta  \int_{\mathbb R^d} |\nabla u_0(x)|^2$ for some  fix $\delta >0$.
 
 
Now, let $I$ be the maximal interval of existence of the above NLS. 

>**My Question**: How to show $\int_{\mathbb R^d} |\nabla u(x, t)|^2 dx \leq (1-\delta) \|\nabla W \|^2_{L^2}$  and $\int_{\mathbb R^d} |\nabla u(x, t)|^2 - |u(x, t)|^{2d/d-2} dx \geq \delta  \int_{\mathbb R^d} |\nabla u(x, t)|^2 dx$
for all $t\in I.$ 
In other words, if these inequalities are true at time $t=0$, then why they are true for all time $t\in  I$?


 (Why this is call energy trapping? Because $E(u(t))\geq 0$?)

[**My confusion:** The author gives the hint that use continuity argument, what is the meaning of continuity arguement and how to apply here.]