Increasing Function or Polynomial with Prescribed Values

Consider $n$ points $(a_1,b_1), (a_2,b_2),\cdots, (a_n,b_n)$ in Euclidean plane with $a_1<a_2<\cdots < a_n$ and $b_1<b_2<\cdots < b_n$. It is easy to construct a polynomial of degree $n-1$ whose graph passes through these $n$ points.

Question: Can we construct a polynomial function $p(x)$ which is increasing with $p(a_i)=b_i$? (I will not put restriction on degree of $p(x)$.) If this is difficult to construct, is is possible to construct a function $f(x)$ which is increasing and $f(a_i)=b_i$?

(Intuitively, it is clear that there exists continuous (or smooth) functions which are increasing and whose graph passes through the $n$ point given above with prescribed order. I am expecting construction.)

• The answer to all your questions are YES. Google "Monotone interpolation" Commented Sep 30, 2014 at 11:29
• By the way, I asked the same question some time ago on MO : mathoverflow.net/questions/16673/… Commented Sep 30, 2014 at 11:30

Yes, at least to the second question: you just construct the piecewise-linear interpolant for the $(a_i, b_i)$ pairs, i.e., you "connect the dots". That's an increasing function that takes on the specified values.