Extending a function from $\Bbb N$ to $\Bbb R$

Question

Consider $f: \Bbb N \to \Bbb N$, a strictly monotonic function (this also means it is increasing).

Now suppose we want to create a function $g: \Bbb R_+ \to \Bbb R_+$, which has the following properties:

1. $g$ and $f$ agree at all $n \in \Bbb N$
2. $g$ is monotonic
3. $g$ is continuous at all points

Can we further restrict 3, for example require it to be infinitely differentiable or analytic, so as to make $g$ exist uniquely?