We call a set right scattered (left scattered) if every nonempty subset contains a point isolated on the right (left). We establish the following monotonicity theorem for the symmetric derivative. If a real function
f
f
has a nonnegative lower symmetric derivate on an open interval
I
I
, then there is a nondecreasing function
g
g
such that
f
(
x
)
>
g
(
x
)
f(x) > g(x)
on a right scattered set and
f
(
x
)
>
g
(
x
)
f(x) > g(x)
on a left scattered set. Furthermore, if
R
R
is any right scattered set and
L
L
is any left scattered set disjoint with
R
R
, then there is a function which is positive on
R
R
, negative on
L
L
, zero otherwise, and which has a zero lower symmetric derivate everywhere. We obtain some consequence including an analogue of the Mean Value Theorem and a new proof of an old theorem of Charzynski.