Using terahertz-light excitation, we have measured with sub-wavelength spatial, and sub-cycle temporal resolution the time- and frequency-dependent electric-field and surface-charge density in the vicinity of small metallic holes. In addition to a singularity like concentration of the electric field near the hole edges, we observe, that holes can act as differential operators whose near-field output is the time-derivative of the incident electric field. Our results confirm the well-known predictions made by Bouwkamp, Philips Res. Rep. 5, 321-332 (1950), and reveal, with unprecedented detail, what physically happens when light passes through a small hole.