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Does the notation $ u \in C^0 ([0,T], H^{s} (\Bbb R^n ) ) $ imply $ \lim_{k \to \infty}\| u(t_k ) - u(t_0 ) \|_{H^{s} (\Bbb R^n )} = 0 \;\;\text{if} \;\; \lim_{k \to \infty} t_k = t_0$ as well as $ \lim_{k \to \infty} \| u(t_k) \|_{H^s (\Bbb R^n)} = \| u(t_0 ) \|_{H^s (\Bbb R^n )} \;?$ Here $t_k$ is a sequence in $[0,T]$ and $H^s$ is Sobolev space ($s=0,1,2,\cdots$).

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Yes. In fact, what you wrote is the definition of the space $C^0([0,T];H^s(\mathbb{R}^n))$. That space consists of all continuous functions $u : [0,T] \to H^s(\mathbb{R}^n)$, where we mean continuity with respect to the $\|\cdot\|_{H^s}$-norm, i.e. your first equation must hold.