Which class of functions have the property of being always elastic?and instead always inelastic?
I don’t think that there are any classes of functions where you can ex ante say that it is elastic or inelastic, since this will depend on the function parameters.
However, there is a class of functions for which you can say the elasticity is always constant, these are isoelastic functions.
For example, consider function:
By definition elasticity for this function is:
Now this kind of function will be always elastic ($b$>1), inelastic ($b$<1) or unit elastic ($b$=1) depending on the actual value of the parameter $b$