  We investigate the statistics of eigenstates in a weak self-affine disordered
potential in one dimension, whose Gaussian fluctuations grow with distance with
a positive Hurst exponent $H$. Typical eigenstates are superlocalized on
samples much larger than a well-defined crossover length, which diverges in the
weak-disorder regime. We present a parallel analytical investigation of the
statistics of these superlocalized states in the discrete and the continuum
formalisms. For the discrete tight-binding model, the effective localization
length decays logarithmically with the sample size, and the logarithm of the
transmission is marginally self-averaging. For the continuum Schr\"odinger
equation, the superlocalization phenomenon has more drastic effects. The
effective localization length decays as a power of the sample length, and the
logarithm of the transmission is fully non-self-averaging.
