Let $\phi \in C^{\infty}(\mathbb{R}^n)$. Consider $\phi(x,y)$ as a function on $\mathbb{R}^p \times \mathbb{R}^q$, where $p+q=n$. For any $y \in \mathbb{R}^q$, consider the map $\phi_y:\mathbb{R}^p \rightarrow \mathbb{R} : x \mapsto \phi(x,y)$. Then do we have $\phi_y \in C^{\infty}(\mathbb{R}^...

Let $a<b<c<d$ be real values, and let $f \in C^{\infty}([a,b])$ and $g \in C^{\infty}([c,d])$. Is there a way to "connect" these functions in a smooth way? That is, is there a function $h \in C^{\infty}([a,d])$ such that $h=f$ on $[a,b]$ and $h=g$ on $[c,d]$? If this is true, how is the proven? ...