Glue Together smooth functions Let's say that $f(x)$ is a $C^{1}$ function defined on a closed interval $I\subset \mathbb{R^{+}}$ and $g(x)\equiv c$ ($c$=constant) on an open interval $J\subset \mathbb{R^{+}}$  where $\overline{J}∩I\neq \emptyset$. Is there a way to "glue" together those two functions in such a way that they connect smoothly? 

If $\overline J \cap I \neq \emptyset$, clearly not. If there is an open interval between $I$ and $J$ then yes, you can interpolate with a polynomial of degree $3$.
