Question: Can we have two real valued functions $f\_1$ and $f\_2$ defined on $[a,b]$ such that $f\_1(x)=f\_2(x)$ for infinitely many points and $f\_1(x)\neq f\_2(x)$ for infinitely many points. ?

Answer: 
$$f\_1(x)=|x|$$
$$f\_2(x)=x$$
where $x\in (-1,1)$.

$\forall x\in [0,1) \ f\_1(x)=f\_2(x)$ but $\forall x \in (-1,0) \ f\_1(x)\neq f\_2(x)$. Both intervals are dense, therefore have infinitely many points.
How about $f\_1(x)=0$ and $f\_2(x)=\begin{cases}x\sin\frac1x &\mbox{if } x\neq 0 \\ 0 &\mbox{if } x=0 \end{cases}$, on the interval $[-1,1]$?

In this case, the functions are both continuous, which is kind of cool.