Can two distinct elementary functions be equal over an interval of nonzero width? The wikipedia entry on elementary functions describes them to be "of a single variable (typically real or complex) that [are] defined as taking sums, products, and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions". Piecewise functions do not fit this description, and I believe these functions are continuous for the regions for which they are defined, as well as their derivatives for the regions along which those derivatives are defined.
Taking two functions which are composed of elementary functions and are unequal for the majority of the interval along which they can be defined for the independent variable, can these functions be equal (meaning they contain all the same points) over an interval with nonzero width, such as being defined and equal for (2,3) or (0, inf)? If this is impossible, why is it impossible?
Thanks!

Extending the ideas already present in the comments, pick any $a<b$.
Let $f(x)=b-a$ and $g(x)=\sqrt{(x-a)^2}+\sqrt{(x-b)^2}$. Both $f(x)$ and $g(x)$ are elementary functions. $g(x)$ is just $|x-a|+|x-b|$ in disguise so that it would be clear that it is an elementary function according to all definitions.
$f(x)$ is identically equal to $g(x)$ for any $x\in[a,b]$.
