  Interactions growing slower than a certain exponential of the square of a
scalar field, are well behaved when evolved under the functional
renormalization group linearised around the Gaussian fixed point. They satisfy
properties usually taken for granted, and reproduce standard perturbative
quantisation. However, ever more challenging effects appear the more
interactions grow faster than this. We show explicitly that firstly the flow no
longer splits uniquely into operators of definite scaling dimension; then
(linearised) flows to the infrared can end prematurely in a singularity; and
finally new interactions can spontaneously appear at any scale.
