Let $a_{n}:\mathbb{N}\rightarrow\mathbb{R}$ a sequence of real numbers. It is easy to see that $a_{n}$ can be extended to a continuous function $a(x):\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ (by connecting the dots on the graph of $a_{n}$ with line seqments) but $a(x)$ is not smooth.
Is there a way to extend the sequence $a_{n}$ to a smooth function $f:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ (I can imagine that this is true)? If yes, is there a way to construct it?