Why do spectra appear continuous despite their allowed transitions being discrete?
You are right that absorption and emission are quantized processes, but UV-visible peaks represent a broad range of wavelengths because there are multiple vibrational states of their representative ground and excited electronic states.



In the case on my diagram, you have six different quantized electronic transitions with slightly different &Delta;E values. In reality, there are more energy levels than that, plus the rotations and vibrations are constantly changing the orbital energy levels as a result of interactions with the matrix, so that helps fill in the peaks. The peak shape can actually be changed by altering the temperature. This gives rise to high-resolution UV-Visible spectrophotometric techniques, which have a few specific applications.