Peeping a bit into the Szekeres-Seminar somewhere on the forum mentioned by Andrew, and rearranging the elementary Schröder functions found by Schröder himself, I can add some more elementary superfunctions:

1. , .

Let us check:

and on the other hand

So its indeed a superfunction.

Edit: It is regular at fixed point :

2. ,

Let us check:

It is again regular at 0.

1. , .

Let us check:

and on the other hand

So its indeed a superfunction.

Edit: It is regular at fixed point :

2. ,

Let us check:

It is again regular at 0.