Let $\; f : [0,1] \to \mathbb{R} \;$ be continuous and non-decreasing. $\;\;$ Let $\epsilon$ be a real number such that $\; 0 < \epsilon \;$.
  

Does it follow that that there exists a real polynomial $p$ such that $p$ is non-decreasing on


1. $\;$ $[0,1]$
2. $\;$ all of $\mathbb{R}$


and for all members $x$ of $[0,1]$, $\; |f(x)+(-(p(x)))| < \epsilon \;\;$?



Stone-Weierstrass for monotone functions
In fact this question is a simple prototype of a serious problem of approximation maintaining additional qualitative properties of a function, with precise error estimates. See 
<http://mathworld.wolfram.com/ComonotoneApproximation.html> 
for the case of piecewise monotone functions. There are many problems and results of this kind (for example, with convex functions), which are useful in engineering applications and are far from trivial. 


