A method has been presented to extract topographical and topological properties of a function defined on an arbitrary set of points in space, strictly via postprocessing with no additional function evaluations. By connecting the points with a neighbourhood graph, welldefined and robust algorithms were developed that allow for identification of local and global maxima (both pointlike and spatially-extended), saddle points, critical paths (and their critical points) and basins of attraction. By simple transformations of the function, one can also identify local and global minima, isosurfaces and stagnation graphs. Applications of the analysis was demonstrated for a few problems in quantum chemistry including Bader charge and bond analysis, identification of valence shells and their charge concentration, location of lone pairs via the electron localization function or the Laplacian of the electron density and identification of stagnation graphs of magnetically-induced currents. All of these investigations were carried out directly on a real-space integration grid used in density functional theory (DFT) calculations, allowing the results to be easily, efficiently and directly incorporated into DFT calculations. The analysis was found to scale as O(N log(N )) where N is the number of grid points and quantities of interest were found to converge rapidly N , requiring orders of tude fewer grid points than uniform-grid methods. Topographical results calculated using such DFT grids were found to exhibit a significant "grid bias" when the algorithm was constrained to stay on the graph. The source of this bias was analysed and found to be removed by allowing off-graph moves. Grant topDFT (grant agreement no. 772259). We are grateful for access to the University of Nottingham High Performance Computing facility.