Prove or disprove that it's possible to approximate the functions in $C([a,b],\mathbb R) $ uniformly with arbitrary error $\epsilon$ by functions in the algebra $\mathcal A$ generated on $[a, b]$ by any nonvanishing strictly monotonic function $\varphi(x)$ . that is to say, $\bar{\mathcal A}\supset C([a,b],\mathbb R)$
Here we may have discontinious $\varphi (x)$.
I have tried for $\varphi(x)\in C([a,b],\mathbb R)$, since $\varphi$ is a homeomorphism, we can consider $C(\varphi [a,b],\mathbb R)=\{g:f\circ\varphi^{-1}|f\in C([a,b],\mathbb R)\}$ hence we can use the weierstrass approximation to find a polynomial $P_n=Q\circ\varphi^{-1} $ to approximate $f\circ\varphi^{-1}$, then use $Q$ to approximate $f$, and show that $Q=P_n\circ\varphi$ is in our $\mathcal A$. I'm not sure if this is Right.
For discontinious $\varphi(x)$, I can't find a possible solution.
Feel free to give any hints or point out mistakes above.