This is not true in general.
For example consider the case $a=0, b=4$, so $R = \Bbb Z[2i]$
$N(c+dx) = |c+2di|^2 = c^2+4d^2$.
$x = 2i$ has norm $4$, $2+x = 2+2i$ has norm $8$, but there is no element of norm $2$ (because that would require $d=0$ and then $c^2=2$)
However, your conjecture is tru...
In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties:
V is differentiable everywhere
The derivative V ′ is bounded everywhere
The derivative is not Riemann-integrable.
== Definition and construction ==
The function is defined by making use of the Smith–Volterra–Cantor set and "copies" of the function defined by
f
(
x
)
=
x
2
sin
(
...
