# Functions with null derivative

I am not here referring to the devil staircase, but to the question mark function. This is a strictly increasing function from $\mathbb{Q}$ to $\mathbb{Q}$, with derivative always $0$. I have two questions:

1. Is this a surprising result (in the same way that continuous functions nowhere differentiable were thought surprising when discovered)?

2. What are the properties preventing this (obviously, we need a topological field to speak of derivative; are there exemples of connected topological fields with non constant functions having everywhere a null derivative?)

• What is a "topological field" and why do we need it to speak of the derivative? – GH from MO Aug 28 '12 at 14:44
• A topological field is a field with a topology such that the operations $x\mapsto -x$ $x,y\mapsto x+y$ $x,y\mapsto xy$ and $x\mapsto x^{-1}$ are continuous. One will need a topology in order to take any sort of limit operation such as a derivative. – 35093731895230467514051 Aug 28 '12 at 15:30
• @Joseph: Thanks for the clarification! – GH from MO Aug 28 '12 at 16:07