Groups are not inherently discrete - there are many continuous groups (called Lie groups) such as the circle, group of rotations in 3D, etc etc. You can also se

Groups are not inherently discrete - there are many continuous groups (called Lie groups) such as the circle, group of rotations in 3D, etc etc. You can also see this turn up in Fourier theory: the discrete Fourier transform has an underlying discrete group, Fourier series for a periodic function are really using the circle group underneath, and the Fourier transform on the real line is using the continuous group of the real line, with addition as the group operation. Elliptic curves are nice because they are algebraic groups, so by plugging in complex or real numbers you get a continuous group, and by plugging in finite fields you get a discrete group.