  We study the problem of super-resolution, where we recover the locations and
weights of non-negative point sources from a few samples of their convolution
with a Gaussian kernel. It has been recently shown that exact recovery is
possible by minimising the total variation norm of the measure. An alternative
practical approach is to solve its dual. In this paper, we study the stability
of solutions with respect to the solutions to the dual problem. In particular,
we establish a relationship between perturbations in the dual variable and the
primal variables around the optimiser. This is achieved by applying a
quantitative version of the implicit function theorem in a non-trivial way.
