Any examples that a function on a subset of a manifold cannot be smoothly extended?

Question

Let $U \subseteq M$ be an open subset of a manifold $M$, and $f \in \mathcal{C}^{\infty}(U)$ be a smooth function on $U$.

1. In general, can we extend $f$ to a smooth function $g \in \mathcal{C}^{\infty}(M)$?

2. However, is there any example of $U$ and $f$, such that $f$ does not extend to a smooth function $g \in \mathcal{C}^{\infty}(M)$? Please give the example.

3. If $supp(f) \subseteq U$ is not closed in $M$, is there any example that $f$ extends by zero to a smooth function $g \in \mathcal{C}^{\infty}(M)$? Please give the example.

Answer 1

The answer to 1. is no. Consider $f(x)=\dfrac1{1-x^2}$ on $U=(-1,1)\subset\Bbb R$. With regard to 3., the definition of support makes it a closed subset, does it not?