  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 shown that exact recovery is possible by
minimising the total variation norm of the measure, and a practical way of
achieve this is by solving the dual problem. In this paper, we study the
stability of solutions with respect to the solutions dual problem, both in the
case of exact measurements and in the case of measurements with additive noise.
In particular, we establish a relationship between perturbations in the dual
variable and perturbations in the primal variable around the optimiser and a
similar relationship between perturbations in the dual variable around the
optimiser and the magnitude of the additive noise in the measurements. Our
analysis is based on a quantitative version of the implicit function theorem.
