Supersymmetry feels like a discrete symmetry to me, since the fermions are turning into bosons, and vice versa. I understand there is an infinitesimal parameter involved in the transformations, but I don't know what it actually determines physically.
Of course the elephant in the room is the elusive nature of Grassmann-odd numbers, cf. this related Phys.SE post and links therein.
A super-charge $Q$ belongs to the super-Poincare algebra and takes bosons into fermions, and vice-versa.
Very oversimplified (i.e. ignoring the Grassmann-nature), $Q$ acts like a raising or lowering operator, which gives it a discrete feel, cf. OP's question. Think e.g. of $su(2)$-irreps with a discrete $m$ quantum number, such as in the isospin symmetry, which is also a continuous symmetry.