  Consider the self-map F of the space of real-valued test functions on the
line which takes a test function f to the test function sending a real number x
to f(f(x))-f(0). We show that F is discontinuous, although its restriction to
the space of functions supported in K is smooth (and thus continuous), for each
compact subset K of the line. More generally, we construct mappings with
analogous pathological properties on spaces of compactly supported smooth
sections in vector bundles over non-compact bases. The results are useful in
infinite-dimensional Lie theory, where they can be used to analyze the precise
direct limit properties of test function groups and groups of compactly
supported diffeomorphisms.
