# Approximation of a continuous function by the polynomial of a continuous function

Prove or disprove that there does not exist a real valued continuous function $g$ on $[0,1]$ with $g (x ) \neq x$ for all $x \in(0,1)$ such that given any $\varepsilon > 0$ and any real valued continuous function $f$ on $[0,1]$ there exist real numbers $a_1, a_2,...,a_n$ (depending only on $f$,$g$ and $\varepsilon$) such that $$|f(x)-\sum_{k=0}^na_k(g(x))^k|<\varepsilon$$for all $x\in[0,1]$.

I believe the statement is false, but I cannot really prove it. The reason I believe it is false is that I should not be able to find a polynomial in a continuous function which approximates EVERY $f$. I feel this is a consequence of Weierstrass Approximation Theorem. Any hint is appreciated.

The first question that came to my mind is: Should $g$ be a polynomial to start with? But even then, how one can proceed is eluding me. I tried to take $f(x)=x^d$ for natural numbers $d$ but nothing comes to my mind.

Well, without the assumption "$g (x ) \neq x$ for all $x \in(0,1)$", putting $g(x)=x$ fits the bill (Weierstrass killin' it).
You can tweak that a little bit and set $g(x)=\frac{x}{2}$. This satisfies all the requirements.
• Thanks, so such a $g$ indeed exists, in fact $g(x)=\dfrac{x}{2}$. I was getting confused with $n$ as I could not make sure whether $n$ is fixed for every $f$ or not. – Landon Carter Apr 13 '15 at 20:01