Absolute continuous functions property. Is it true?

Question

Let $f,g:[0,T]\to [0,\infty)$ be two absolute continuous functions such that: $f(0)=g(0)>0$. We know that there is a sequence $(x_n)_n$ converging to $0$ such that:

$$f(x_n)\neq g(x_n)$$

Can we infer that there is $\varepsilon>0$ such that:

$$f(x)>g(x),\ \forall\ x\in (0,\varepsilon)\;?$$

or

$$f(x)<g(x),\ \forall\ x\in (0,\varepsilon)\;?$$

Answer 1

I guess no, let $f(x)=x^3\sin\frac 1x$ and $g(x)=0$.