I am exploring the continuity of tangent and normal cone mappings to a nonempty compact set $X$ at $x\in X$; that is, let $T(x;X)$ and $N(x;X)$ respectively denote the tangent and normal cone to $X$ at $x$. $T(x^k;X)\to T(x^*;X)$ and $N(x^k;X)\to N(x^*; X)$ as $x^k\to x^*$.
Is it generally guaranteed? I could not find any results discussing the continuity.
The motivation of the question is to identify the continuity of the sum of a continuous mapping $\Phi\colon\Bbb R^n\to\Bbb R^n$ and the normal cone $N(x;X)$, i.e., $\Phi(x)+N(x;X)$.