contraction mapping coeficient

Question

I have a question concerning the definition of a contraction mapping.

I have found in several books different definitions for the interval from which the constant of contraction is chosen. Some say $c \in [0,1)$ and others say $c \in (0,1)$, without any other additional conditions. Are these two definitions equivalent?

Answer 1

When $c=0$, we have that $d(f(x),f(y))=0$ for all $x$ and $y$. In this case, $f(x)=f(y)$ for all $x$ and $y$, so $f$ is constant. Not particularly interesting.

Answer 2

$c=0$ is the trivial case, right?

It works irrespective of the space and the distance metrics.