I have a compact subset $A$ of $\mathbb{R}^N$ and a $C^1$ function; $f:A\rightarrow \mathbb{R}$. Let $B\supseteq A$ be a compact subset of $\mathbb{R}^N$. Can I find a $C^1$ function $\tilde{f}:B\rightarrow \mathbb{R}$ such that $\tilde{f}|_A\equiv f$? If it is not possible, what is a good counter-example?
I feel like the answer should be that it is possible, but from what I've read online I'm not so sure. I saw that Whitney's approximation theorem might give me something similar to what I need, but I am a little confused by the notions of differentiability used (I was looking at these lecture notes for instance and can't quite tell if Chapter 2 is saying I need extra conditions or not).