[flebron]

Consider a disconnected domain (say, union of a few open balls in R^n), and f being constant in each connected component, but having different values in each ball. The differential is indeed everywhere 0 in the entire domain.

[dataflow]

D'oh, I missed the possibility of an open domain. Thanks!

[sannee]

And, even better, the dimension of the kernel of d just counts how many connected components the domain has.

There is a similar thing in graph theory, where the kernel of the incidence matrix counts how many connected components a graph has.

[bmitc]

Yea, the statement is missing "locally constant". It's a poor example given the assumptions are not clearly stated.

[pfortuny]

Oh, these kind of problems happen time and again in real-life maths: you use a lemma until someone points out that you are assuming something which may not take place (like connectedness of the domain, here).