Band-limited functions can oscillate arbitrarily faster than their fastest Fourier component over arbitrarily long intervals. Where such "superoscillati​ons" occur, functions are exponentially weak. In typical monochromatic optical fields, substantial fractions of the domain (one-third in two dimensions) are superoscillator​y. Superoscillatio​ns have implications for signal processing, and raise the possibility of sub-wavelength resolution microscopy without evanescent waves. In quantum mechanics, superoscillatio​ns correspond to weak measurements, suggesting weak values of observables (e.g photon momenta) far outside the range represented in the quantum state. A weak measurement of neutrino speed could lead to a superluminal result without violating causality, but the effect is too small to explain the speed claimed in a recent experiment.