It is easy to see from the construction of Bernstein polynomials that if $f$ is Lipschitz continuous on $[0,1]$ then the distance of $f$ from the subspace of polynomials of degree at most $n$ is $O(1/n)$ where the implied constant depends on the Lipschitz constant of $f$. More generally any information on the modulus of continuity of $f$ can be used to get some kind of bound on this distance. Now my question is, can one prove any (positive or negative) statement of this kind for arbitrary continuous functions? Is it possible to prove that if a continuous function can be very well approximated by polynomials in some sense then it has to have a property similar to Lipschitz? 



Quantitative Weierstraß approximation
The answer is yes. The results which allow to infer the modulus of continuity of a function from the value of function's best approximation by polynomials are collectively known as Bernstein-type theorems.



> 
> **Theorem** (S.N. Bernstein). Let $E\_n(f)$ be the best approximation to the function $f(.) \in C([a, b])$ 
>  by algebraic polynomials of degree at most $n$, and let
>  $$E\_n(f)=O(n^{-\alpha})$$
>  with some $0< \alpha \leq1$. 
> 
> 
> * If $\alpha < 1$, then $f(.)$ is uniformly Hölder continuous with exponent $\alpha$ on each segment $[a', b'] \subset(a, b)$, i.e.
>  $$\omega(f,\delta)=O(n^{-\alpha}).$$
> * If $\alpha=1$, then $f(.)$ is uniformly almost Lipschitz on each such segment, i.e.
>  $$\omega(f,\delta)=O\left(\delta\ln \frac{1}{\delta}\right).$$
> 
> 
> 


The theorem cannot be improved in the sense that $E\_n(f)=O(n^{-1})$ does not imply that $f(.)$ is Lipschitz. This is somewhat easier to see in case of the essentially equivalent problem of the best approximation of periodic functions by trigonometric polynomials on $\mathbb R$. The modulus of continuity of the Weierstrass function 
$$f\_0(x)=\sum\_{k=1}^{\infty}\frac{\cos(3^kx)}{3^k} $$
is $M\delta\ln\frac{1}{\delta}$. At the same time, one can show that the best approximation of $f\_{0}(.)$ by trigonometric polynomials of degree $\leq n$ is equal to
$$E\_{n}^{Т}(f\_0)=\frac{3}{2}\frac{1}{n}.$$




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A very good reference on Bernstein-type theorems and related results is [*Theory of Approximation*](http://rads.stackoverflow.com/amzn/click/0486671291) by Akhiezer.


