Let $A$ be a finite (and small-ish) set of positive real numbers and 0. Let $B$ be a subset of $\mathbb N^0$, up to some (small-ish) bound.
I have a function $f(t)$, $A \rightarrow B$ that is strictly monotonically increasing. It is $f(0) = 0$ and $f$ shall cover all values of its codomain $B$.
In plain English: $f$ starts at (0,0) and always increases by one, but with varying $\Delta t$ in between the steps.
A simple continuous version of $f(t)$ is $c(t)$ which I can get by piecewise linear interpolation. However since my $\Delta t$ values are all different, the derivative $c'(t) = dc/dt$ will have discontinuities.
I am looking for the easiest possible function $g(t)$ (stretching and squeezing $t$) such that $\frac {dc(g(t))}{dt}$ is smooth, with $g(t)=t$ $\forall t \in A$.
In plain English: How can I turn my simple linear interpolation into something that has a continuous time derivative (but still perfectly interpolates $f$)?