I am trying to understand what is meant by a singular support of a distribution:
So singular support is defined as the complement of the largest open set on which T (the distribution) fails to be a smooth function. My question is how can a distribution not be $C^\infty$ in the first place? By definition of the distributional derivative, we apply the derivative to the test functions which are all $C^\infty$ so it is impossible for a distribution not to be infinitely differentiable.
A distribution $\phi$ is said to be a smooth function if there exists an actual smooth function $f$ satisfying $\phi(g)=\int f g dx$ for all test functions $g$. This is said to hold on an open set $U$ if the equality holds for all test functions with compact support in $U$.