Slightly equal functions

Can there exist two elementary functions $f(x)$ and $g(x)$ defined everywhere on the real axis such that,
\begin{align} f(x)&=g(x)\qquad \text{if} \quad a\le x\le b\\
f(x)&\neq g(x)\qquad \text{if} \quad xb\end{align}
where f(x) and g(x) are not piecewise defined functions. And $a\ne b$.

If yes, give example. If no, give proof.

Also, would it make any difference if the functions need not be elementary?

Edit : It seems there is a lot of confusion due to my inability of putting the question precisely. Please refer to the links.
Elementary functions http://en.wikipedia.org/wiki/Elementary_function
Piecewise defined function http://en.wikipedia.org/wiki/Piecewise

I have also added the 'defined everywhere' condition.