Hi Bridget. I’ve been trying to find out the answer to this for you, but with not a lot of luck. as you go up the spectrum, frequency increases and the energy of the photons increases. Is there a limit to how much energy a photon can have? I can’t find a straight answer – there may be one that Sara can point out.
In any case, by definition the electromagnetic spectrum ends at gamma waves, because the “gamma wave” part of the spectrum is open ended. Anything above a certain energy (below a certain wavelength) gets called a gamma ray. They cover quite a large range of energies, so we really should break them up into a couple of different names at least.
I think it is because this is just what we are calling the longest and shortest waves we can measure at the moment although technically the long wavelength limit is the size of the universe itself and the short wavelength limit is generally thought to be the vicinity of the Planck length.
But I think that you are right in principle the spectrum is infinite and continuous! 🙂