We report on continuous high-harmonic generation (HHG) at 1 kHz
repetition rate from a liquid-sheet plasma mirror driven by
relativistic-intensity near-single-cycle light transients. Through
precise control of both the surface plasma density gradient and the
driving light waveform, we can produce highly stable and reproducible
extreme ultraviolet spectral quasi-continua, expected to correspond to
the generation of stable kHz-trains of isolated attosecond pulses in
the time domain. This confirms the exciting potential of liquid-sheet
targets as one of the building blocks of future high-power attosecond
lasers.