Tetration base 1 is a constant function equal to 1, for x>-1, and has a discontinuity at -1. Or at least that's the solution if .

The oscillatory behavior of bases 0<b<1 suggest that base 0 looks like this on the limit for b=0:

We don't know if tetration has changes of state as the base converges to zero, but can only be equal to 0 or 1, so the doubt is how many discontinuities it has on a given interval.

Base -1:

, so, once , then the function must be periodically equal to -1:

This is a possible solution:

Is not the only solution; it may be displaced to the right by any real value d (0<d<1), and still will be a solution valid on the interval -2+d<x.

I wonder if there is a family of continuous solutions, and this one is the evolvent.

The oscillatory behavior of bases 0<b<1 suggest that base 0 looks like this on the limit for b=0:

We don't know if tetration has changes of state as the base converges to zero, but can only be equal to 0 or 1, so the doubt is how many discontinuities it has on a given interval.

Base -1:

, so, once , then the function must be periodically equal to -1:

This is a possible solution:

Is not the only solution; it may be displaced to the right by any real value d (0<d<1), and still will be a solution valid on the interval -2+d<x.

I wonder if there is a family of continuous solutions, and this one is the evolvent.

I have the result, but I do not yet know how to get it.