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*A blackbody is also a perfect emitter giving off electromagnetic waves at all frequencies. A detector could measure the intensity of the radiation it receives through the prism. By moving the detector to different positions, you could measure the intensity of light as a function of color or wavelength.*(https://goo.gl/bqwl7K)

So each wavelength has a finite (not an infinitesimal) intensity? If we consider wavelengths in the interval [λ, λ + dλ] and each wavelength in this interval has a finite intensity, then the total intensity for this interval would be infinite because this interval has infinitely many wavelengths. Where am I going wrong? Are there not infinitely many wavelengths within this interval?

Between the red and green beams there are infinitely many beams with a finite intensity, so the total for this interval would be infinite?