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: 
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.
Consider
$$f\_1:[a,b]\to \mathbb{R}:x \mapsto \left\{\begin{array}{l l} 1 & \text{if } a \leq x \leq \frac{a+b}{2} \\ 0 & \text{else}\end{array}\right.$$
and $$f\_2:[a,b]\to \mathbb{R}:x \mapsto \left\{\begin{array}{l l} 1 & \text{if } a \leq x \leq \frac{a+b}{2} \\ 2 & \text{else}\end{array}\right.$$
then $f\_1 ([a,\frac{a+b}{2}])= f\_2 ([a,\frac{a+b}{2}])$ and $f\_1((\frac{a+b}{2},b]) \neq f\_2((\frac{a+b}{2},b])$.