Consider $C^{\infty}(U, \mathbb{R}^{k})$ and $C^{\infty}(V, \mathbb{R}^{k})$, where $V\subset U $, with $U$ and $V$ open subsets of $\mathbb{R}^n$. Is it true that the restriction
\begin{align} R:C^{\infty}(U, \mathbb{R}^{k}) &\rightarrow C^{\infty}(V, \mathbb{R}^{k})\\ f &\mapsto f \big|_{V}\end{align} is continuous?
It's certainly not surjective. Concoct your own example; say $f\in C^\infty(V)$ and $f$ blows up at a boundary point of $V$...
I'm not sure what the "Whitney" topology is. The standard topology on $C^\infty(U)$ is metrizable, with $f_n\to f$ if and only if $D^\alpha f_n\to D^\alpha f$ uniformly on every compact subset of $U$. With this topology it's trivial from the definition that the restriction is continuous. (Say $f_n\to f$ in $C^\infty(U)$. Let $K$ be a compact subset of $V$. Then $K$ is a compact subset of $U$, so...)