behavior of function between two bounds

Question

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 $[0,x_0)$?

Answer 1

No. Take $L(x)=x^2$, $U(x)=x$, $f(x)=L(x)+\mathbf 1_{\mathbb Q}(x)\cdot (U(x)-L(x))=\begin{cases}U(x)&\text{if $x\in\mathbb Q$,}\\L(x)&\text{if $x\notin\mathbb Q$.}\end{cases}$

Answer 2

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