  We uncover how nonlinearities dramatically alter the buckling of elastic
beams. First, we show experimentally that sufficiently wide ordinary elastic
beams and specifically designed metabeams ---beams made from a mechanical
metamaterial--- exhibit discontinuous buckling, an unstable form of buckling
where the post-buckling stiffness is negative. Then we use simulations to
uncover the crucial role of nonlinearities, and show that beams made from
increasingly nonlinear materials exhibit increasingly negative post-buckling
slope. Finally, we demonstrate that for sufficiently strong nonlinearity, we
can observe discontinuous buckling for metabeams as slender as $1\%$
numerically and $5\%$ experimentally.
