$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$ $\newcommand{\sgn}{\operatorname{sgn}}$ $\newcommand{\rank}{\operatorname{rank}}$
Consider matrix $A(w)$ depending on the parameter $w$. If eigenvalues are simple then eigenvalues depend on $w$ smoothly or analytically (depending on how matrix $A(w)$ depends on $w$) and the same is true for eigenvectors.
If eigenvalues are not simple then even the Jordan structure of matrix $A(w)$ is not necessarily constant: f.e. $A= \begin{pmatrix} 0 &1 \\\ 0 &w\end{pmatrix}$ for $w=0$ has one Jordan cell of dimension $2$ and for $w\ne 0$ has two simple eigenvalues.
If we know that the roots of characteristic polynomial have constant multiplicities then eigenvalues depend on $w$ smoothly or analytically (depending on how matrix $A(w)$ depends on $w$) but the Jordan structure of matrix $A(w)$ is not necessarily constant: f.e. $A= \begin{pmatrix} 0 &w \\\ 0 &0\end{pmatrix}$.
Consider Hermitean matrix $A(w)$ depending on the parameter $w\in \bR^d$, $d>1$. Then eigenvalues have first-order directional derivatives bounded but eigenvectors are not necessarily continuous. F.e. $A= \begin{pmatrix} w_1 &w_2 \\\ w_2 &-w_1\end{pmatrix}$ or $A= \begin{pmatrix} w_1 &w_2+iw_3 \\\ w_2-iw_3 &-w_1\end{pmatrix}$.
I am not sure about $d=1$.