Can a discrete function converge to a continous function?

Question

Let $f\in C^{\infty}[a,b]$, let also $X \subset [a,b] = \left\{x_0,\ldots,x_k \right\}, Y = \left\{ f(x_0),\ldots, f(x_k) \right\}$. I guess that if I let $k\rightarrow \infty$ then some how I should be able to retrieve $f$ Is it possible to prove that? or the only way is to build an approximation by such samples an d then prove that such approximation converges?

Answer 1

If $f$ is continuous, and thus in particular uniformly continuous on the closed interval $[a,b]$, then making successively tighter approximations (e.g. by linear interpolation) should give you functions that get closer and closer to $f$, and thus you are able to "retrieve" it. Note, that it is not sufficient that $k \to \infty$, you actually need that $$ \lim_{k\to\infty}\max_{0 \leq i < k} |x_i - x_{i+1}| = 0. $$ Otherwise you could pick all points $x_i$ in, say, $[\frac{a+b}2,b]$, and never properly approximate $f$ on the first half of the interval.