Why is the graph of fluorescence vs. wavelength continuous?
There are two main reasons for what you're describing. 

The first is the fundamental properties of the molecules involved, as mentioned in the comments. Even for isolated molecules, the transition is not always between the same vibrational states of the excited and ground state. Since the vibrational states are quantized, we would expect a series of very closely spaced emission lines, each representing a different combination of vibrational states. 

If the molecules are in a solution or in liquid or solid phase, then the interactions of the emitting molecule with the molecules around it can also cause small changes in the energy of the emission as the electronic structure of the molecule is perturbed. 

The general term for these phenomena is "line broadening," and most reference books about spectroscopy will have a discussion of the various mechanisms that lead to it.

The second reason for it is the error in measurements. Most spectrometers cannot filter wavelengths perfectly, so, for example, light of wavelength 590 nm will contribute to a signal at 589.5 and 591.5 nm, less as you get farther away. Sometimes the slit width is an adjustable parameter, but even at the narrowest setting, most instruments will not give extremely high resolution. As a result, a series of closely spaced lines will appear instead as a smooth curve.