I considered an elliptic evolutionary problem and obtained
$$\|u\|_{C(0,T,V)}+\|u'\|_{\mathcal{W}}\leq C(1+\|u(0)\|_V+\|u'(0)\|_{V}+\|f\|_{V^*}),\qquad (1)$$ for all $t\in [0,T]$ and some $C>0$. I started to think what does that mean to have the property of continuous depandance on the data? So I took a sequance of initial conditions $(u_n(0))$ and $(u'_{n}(0))$ both convergent (with respect to their norms) to $u(0)$ and $u'(0)$, respectively. Then, from (1), associated sequances $(u_n)$ and $(u'_n)$ tend to $u$ and $u'$, respectively, which I guess explains the "continuity".
Is there any deeper idea which stands behind (1)? Where is it really needed - in control theory, maybe?