behavior of function between two bounds Let $f, U, L : [0,1] \rightarrow \mathbb{R}$ be three functions with the property that
(1) U and L are continuous functions
(2) $\forall x \in [0,1]$, $L(x) \leq f(x) \leq U(x)$
(3) $f(0)=L(0)=U(0)=C$
(4) $L(x)$ and $U(x)$ are increasing.
We do not know whether $f$ is continuous. 
Do these properties imply that there exists $x_0>0$ such that $f(x)$ is continuous and increasing in $[0,x_0)$?

No Let $L(x)=x,U(x)=2x$, $f(x)=\frac{3}{2}x$ on rational number and $f(x)=\frac{4}{3}x$ on irrational number 
