To summarize my answer quickly:
In our current model of physics, there are plenty of fundamental* things which are actually continuous. Whether the fundamental things remain continuous as science makes progress, nobody can say.
Now it's time to substantiate that statement. All it takes, though, is to find any variable which we model as continuous, and doesn't have an accepted**, more fundamental, discrete origin. There is not only one, there are many:
- Time, as a parameter
- Distance (x,y,z), as a parameter
- The real and complex parts of the wave function at any given point; any spinor component thereof
- The eigenfunctions of field eigenstates
- Possible outcomes of measurements of position, momentum, energy, and other variables, depending on the particular system (e.g. not including bound states).
This list could be extended but since one was enough I'll stop here.
*What is and isn't fundamental isn't always a question which is answerable by the scientific method. I have used my own (not absurdly uncommon) notions of the meaning of that word.
**I will not consider explanations which are not accepted by the scientific community at this time. Not accepted $ \neq $ false, but there are too many alternatives and most are at best unverified.
How does this mesh with the seemingly opposite answer of Vladimir Kalitvianski? His current answer (paraphrased) is that single photons are discrete, as are any finite number of photon measurements. This is correct but it does not imply that all variables are fundamentally continuous; it only talks about photons in particular. From an experimental perspective, we have variables which at least seem to be continuous, in the sense that at all measured resolutions we have achieved, they were not discontinuous (take the double slit for positions, for example). This allows for the possibility for a number of continuous variables, up to our current knowledge / precision.
