Extensions of diffeomorphisms of $S^2$ and the connectedness of $\text{Diff}^+(S^2)$ In this MO question by Daniel Moskovich, he claims that the fact that every diffeomorphism of $S^2$ extends to a diffeomorphism of $D^3$ implies that $\text{Diff}^+(S^2)$, the group of orientation-preserving diffeomorphisms of the 2-sphere) is connected. 
Why does this follow? It doesn't seem obvious to me. 

Any orientation-preserving diffeomorphism $f$ of a star-shaped subset of $\Bbb R^n$ is isotopic to the identity. To see this, show $f$ is isotopic to $df_0$ (assuming the region is star-shaped with respect to $0$ and $f(0)=0$), and then it's a standard linear algebra exercise that $GL(n,\Bbb R)^+$ is connected.
To see the first step, you want to use the usual $C^\infty$ trick and write $$f(x)=\sum x_i g_i(x), \quad\text{where } g(0) = df_0.$$
