Zero derivative implies locally constant
From Calculus
Statement
For an interval in the domain
Suppose is a function and
is an interval (possibly open, closed, or half-open and half-closed) contained inside the domain of
such that
for all
in the interior of
and
is continuous on all of
(note that we do not require differentiability, or even one-sided differentiability, at the endpoints of
, if any). Then,
is a constant function on all of
, i.e. there is a real number
such that
for all
.