A natural way to measure the nonlinearity of a function , where
is an interval, is the quantity
which expresses the deviation of
from a line, divided by the size of interval
. This quantity was considered in Measuring nonlinearity and reducing it.
Let us write where the supremum is taken over all intervals
in the domain of definition of
. What functions have finite
? Every Lipschitz function does, as was noted previously:
. But the converse is not true: for example,
is finite for the non-Lipschitz function
, where
.

The function looks nice, but is clearly unbounded. What makes
finite? Note the scale-invariant feature of NL: for any
the scaled function
satisfies
, and more precisely
. On the other hand, our function has a curious scaling property
where the linear term
does not affect NL at all. This means that it suffices to bound
for intervals
of unit length. The plot of
shows that not much deviation from the secant line happens on such intervals, so I will not bother with estimates.
The class of functions with
is precisely the Zygmund class
defined by the property
with
independent of
. Indeed, since the second-order difference
is unchanged by adding an affine function to
, we can replace
by
with suitable
and use the triangle inequality to obtain
where . Conversely, suppose that
. Given an interval
, subtract an affine function from
to ensure
. We may assume
attains its maximum on
at a point
. Applying the definition of
with
and
, we get
, hence
. This shows
. The upshot is that
is equivalent to the Zygmund seminorm of
(i.e., the smallest possible M in the definition of
).
A function in may be nowhere differentiable: it is not difficult to construct
so that
is bounded between two positive constants. The situation is different for the small Zygmund class
whose definition requires that
as
. A function
is differentiable at any point of local extremum, since the condition
forces its graph to be tangent to the horizontal line through the point of extremum. Given any two points
we can subtract the secant line from
and thus create a point of local extremum between
and
. It follows that
is differentiable on a dense set of points.
The definitions of and
apply equally well to complex-valued functions, or vector-valued functions. But there is a notable difference in the differentiability properties: a complex-valued function of class
may be nowhere differentiable [Ullrich, 1993]. Put another way, two real-valued functions in
need not have a common point of differentiability. This sort of thing does not often happen in analysis, where the existence of points of “good” behavior is usually based on the prevalence of such points in some sense, and therefore a finite collection of functions is expected to have common points of good behavior.
The key lemma in Ullrich’s paper provides a real-valued VMO function that has infinite limit at every point of a given set
of measure zero. Although this is a result of real analysis, the proof is complex-analytic in nature and involves a conformal mapping. It would be interesting to see a “real” proof of this lemma. Since the antiderivative of a VMO function belongs to
, the lemma yields a function
that is not differentiable at any point of
. Consider the lacunary series
. One theorem of Zygmund shows that
when
, while another shows that
is almost nowhere differentiable when
. It remains to apply the lemma to get a function
that is not differentiable at any point where
is differentiable.