
The Glue Function

This denizen of counterexample space is important in calculus on
manifolds. However, that's not why this page is here.
Rather, the page is here because I think this function is so incredibly
cool. This is a liesurely stroll through a few of the glue
function's properties, just because I think they're neat.
The Basic Function
Consider the exponential function, e
^{x}. As x >
infinity, the exponential goes smoothly to zero. Furthermore, it
goes to zero faster than any polynomial goes to infinity  any
polynomial, times e
^{x}, also goes to zero as x>infinity.
The exponential function is the "magic ingredient" in the glue function.
Next, let's squish the whole number line down into the interval
(1,+1), so that our function goes to zero at a finite location.
For a "squish function", we can use s(x) = 1/(1  x
^{2}).
If we just compose f(x) with s(x), we obtain
which looks like this:
Plot 1:
(2) is infinitely differentiable in the interval (1,1). If we
differentiate it, we find something remarkable: All of its
derivatives go to
zero at the end points. As x approaches
either +1 or 1, the derivative with respect to x of g(s(x)) approaches
zero.
This is remarkable because it means we can "piece" this with a constant
function, to obtain:
and h(x) is infinitely differentiable everywhere.
Some Properties
The first odd thing about this function is that, because it's
infinitely differentiable everywhere, we can find a Taylor series
approximation to it at any point. And for points within (1,+1)
the Taylor series will actually converge to the function, just as we
expect  or at least, it will converge to it in a neighborhood of the
point where we construct it.
For points
outside the interval [1,+1], the Taylor series is
just the constant function, 0  the series gives us no hint that
there's any bump in the function, anywhere.
And for +/ 1, the radius of convergence of the Taylor series is
zero!
For an apparently wellbehaved function, this is most strange.
If we look at the derivatives of the function, it becomes clear what's
going on. Each derivative tends to zero as x goes to +/ 1, and
each derivative is bounded.
But the bounds get wider and
wider as the the order of the derivatives gets higher. Looking at
all the derivatives together, they're unbounded as we approach the
endpoints  they are progressively worse behaved as their order gets
higher. In
consequence, if we try to evaluate the Taylor series at +/ 1, no
matter how small we make the neighborhood, there will be derivatives in
it which are arbitrarily large  and the Taylor series won't converge
to the function, as a result.
[Some actual graphs of several of the derivatives should appear
here, eventually]
The Final Step: Making Glue
There's one last trick to making something we can use to stick things
together: We need a step function.
Let's
integrate what we've got so far. The
derivative of the integral is just the function, so the result will
still be infinitely differentiabl everywhere  but instead of being
zero almost everywhere, with a bump in the middle, it will start at
zero, then suddenly step up to a new value, then remain at that value:
With we obtain:
Plot 2  H(x):
And for the last step, we'll normalize it, so that the function rises
from 0 to 1 on the interval (0,1):
which is finally what we want:
Plot 3  g(x) = H(2x  1)/H(1):
By a simple change of variable and by scaling, we can use g(x) to build
a function that goes from any constant value to any other constant
value, on any interval. By multiplying two copies together with
one of them reversed (stepping down), we can build a function that's
constant on any interval we like, and zero everywhere outside some
(larger) interval. We can make it go from 0 to any constant value
in as short a distance as we care to.
And it's always infinitely differentiable, everywhere.
Applications for the Glue Function
The most important application is surely the creation of "
partitions of unity",
which are used throughout the study of calculus on manifolds in order
to stitch together objects that are defined locally into global objects
covering the manifold.
Generally, the glue function can be used to stick together any
differentiable functions that would otherwise not "match". Build
a glue function that's 1 on the domain where the function is defined,
and 0 outside it (with a small "frame" in which it can fall from 1 to
zero), and multiply your original function by the glue.
A completely smooth notquitehalfparabola, for instance, could be
built as
x
^{2} * g(1000*x)
It's an exact parabola for values of x >= 0.001, it's exactly zero
for all values of x <= 0, and it's smoothly joined in the interval
(0,0.001).
More to follow...
Some graphs on this page were not done with Gnuplot, because
Gnuplot can't plot integrals that are not available in closed
form. An old "solver" package I had lying around was used instead.