In Wiki the definition of T is a continious distribution given by :$T$ is continuous distribution if and only if for every compact subset $K$ of $ U $ there exists a positive constant $C_K$ and a non-negative integer $N_K$ such that:
$|T(\phi)|\leq C_K\sup{|{\partial} ^{\alpha}\phi(x)| | x\in K,|\alpha|\leq N_K}$ with ${\alpha}$ is Multi-index , Now my question here is : Why we should have $ |\alpha|\leq N_K$ to get a continuous distribution ?
This is exactly a somewhat formulaic description of the topology on test functions $\mathcal D$, (a strict colimit of Frechet spaces $\mathcal D_K$, test functions supported on compacts $K$). The colimit property exactly is that a continuous linear map $T:\mathcal D\to V$ for some other (locally convex) topological vector space is given by a (compatible in the obvious way) collection of continuous linear maps $T_K:\mathcal D_K\to V$.
Thus, the issue of continuity is exactly about continuity on the Frechet "limitands" $\mathcal D_K$. These are (projective) limits of Banach spaces, namely, the completions of $\mathcal D_K$ with respect to the norms $\nu_N(f)=\sup_{x\in K}\sup_{|\alpha|\le N} |f^{(\alpha)}(x)|$. Why the latter? Because $C^{(N)}(K)$ is indeed complete, with this norm. That is, it is the "correct" topology to put on $C^{(N)}(K)$, and inevitably entails the (Frechet) limit topology on $C^\infty(K)$, etc.