The effects of exposure to ionizing radiation are central in many areas of science and technology, including medicine and biology. Absorption of UV and soft-x-ray photons releases photoelectrons, followed by a cascade of lower energy secondary electrons with energies down to 0 eV. While these low energy electrons give rise to most chemical and physical changes, their interactions with soft materials are not well studied or understood. Here, we use a low energy electron microscope to expose thin organic resist films to electrons in the range 0-50 eV, and to analyze the energy distribution of electrons returned to the vacuum. We observe surface charging that depends strongly and nonlinearly on electron energy and electron beam current, abruptly switching sign during exposure. Charging can even be sufficiently severe to induce dielectric breakdown across the film. We provide a simple but comprehensive theoretical description of these phenomena, identifying the presence of a cusp catastrophe to explain the sudden switching phenomena seen in the experiments. Surprisingly, the films undergo changes at all incident electron energies, starting at ∼0 eV.