Let $U$ be a non empty open subset of $\mathbb{R}^n$ ($n \geq 1)$. I do not assume any other properties on this set (not necessarily bounded, connected, etc.).
Is the following property always true ?
$$ \left.\begin{array}{l} C^1(U) \ni f_k \rightarrow f \text{ in } C^0(U), \\ Df_k \rightarrow g \text{ in } C^0(U), \\ \end{array}\right\} \Longrightarrow f \in C^1(U) \text{ and } Df=g, $$
(the functions $f_k,f$ are real-valued).
For instance, it is true for connected $U$. What if it is not connected ?
The conclusion is a 'local property'. You only have to show that the conclusion holds at each point. Take an open disk around the point contained in $U$ and apply the result to the disk.