Despite Planck's constant being in $E=hf$, it would appear to me that energy is still not discrete, as frequency can be an fraction of a Hertz that one wants. How does this imply that electromagnetic radiation is quantized?
In my books, the energy content of light is not quantized generally in the same sense that the energy of an electron is not quantized generally. The electron can have a continuous energy spectrum as all other free particles.
It is only particles bound in potentials that get quantized energies, and then the photons that mediate transitions between these energy states are quantized. Thus we get the emission and absorption spectra of atoms.
emission spectrum of iron
In this sense these particular photons are quantized in energy.