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THz spectroscopy is an emerging tool for detection of microorganisms and harmful compounds in the food industry, the study of proteins in biomedicine and the development of electron-beam X-ray sources for molecular imaging and lithography. Recording of THz electric field evolution in single-shot is crucially needed in terahertz spectroscopy of irreversible processes in such applications as well as for data communication in the THz portion of the spectrum where there is an abundance of untapped bandwidth. However, achieving sub-picosecond resolution over a long time window has been an open problem for electro-optic sampling -- the standard technique for recording terahertz waveforms. We introduce a new conceptual framework for this open problem that is inspired by time-stretch theory. The novel framework unveils a solution to this 20 year-old problem leading to a dramatic enhancement of the achievable temporal resolution. We validate this new technology in two applications. First, we present single shot recordings of long free-propagating terahertz transients with record time resolution. Second, we present recordings of ultra-short relativistic electron bunches at the European X-ray Free Electron Laser. These results show that electric signals may be now recorded with terahertz bandwidth over arbitrarily long windows, thus enabling the realization of "single-shot terahertz oscilloscopes" and single-shot time-domain spectroscopy systems with an arbitrary time-bandwidth product.