5.3.1 One-sided differentials

Let
be a continuous function (nothing beyond continuity is required from
).
A vector
is called a *super-differential* of
at a given point
if for all
near
we have the relation

Geometrically, is a super-differential if the graph of the linear function , which has as its gradient and takes the value at , lies above the graph of at least locally near (or is tangent to the graph of at ). Figure 5.5(a) illustrates this situation for the scalar case ( ), in which is the slope of the line. A super-differential is in general not unique; we thus have a

Similarly, we say that
is a *sub-differential* of
at
if

The graph of the linear function with gradient touching the graph of at must now lie

We now establish some useful properties of super- and sub-differentials.

TEST FUNCTIONS.
if and only if there exists a
function
such that
,
, and
for all
near
, i.e.,
has a
local minimum at
. Similarly,
if and only if there exists a
function
such that
and
has a
local *maximum* at
. (Note that we can always arrange to have
by adding a constant to
, which does not affect the other conditions.)

The function
is sometimes called a *test function*. For the case of
, an example of such a function is shown in Figure 5.7. The above result, whose proof we will not give, will be used for proving the other facts that follow.

RELATION WITH CLASSICAL DIFFERENTIALS. If is differentiable at , then

If and are both nonempty, then is differentiable at and the relation (5.33) holds.

We prove both claims with the help of test functions. First, suppose that is differentiable at . It is clear that the gradient is both a super-differential and a sub-differential of at (indeed, with both (5.31) and (5.32) become equalities). If has a local minimum or maximum at for some , then hence . This shows that is the only element in and .

To prove the second claim, let be such that and for near . Then has a local maximum at , implying that hence . For (the case is completely analogous) we can write

As , the first fraction approaches , the last one approaches , and we know that the two gradients are equal. Therefore, by the ``Sandwich Theorem" the limit of the middle fraction exists and equals the other two. This limit must be , and everything is proved.

NON-EMPTINESS AND DENSENESS. The sets and are both nonempty, and actually dense in the domain of .

The idea of the proof is sketched in Figure 5.8. The highly irregular graph in the figure is that of . Take an arbitrary point . Choosing a function steep enough (see the solid curve in the figure), we can force to have a local maximum at a nearby point , as close to as we want. (For clarity, we also shift the graph of vertically to produce the dotted curve in the figure which touches the graph of at .) We then have . A similar argument works for .