Are there any true discontinuities in physics? When we first learn physics, it's often presented very 'discontinuously'. For example, pop quantum likes to talk about objects being "either" particles or waves, leading to a lot of confused questions about how things switch between the two. Once you learn about wavefunctions, the problem goes away; 'particle' and 'wave' are just descriptions of two extreme kinds of wavefunctions.
In general, further learning 'fills in' the knowledge holes that discontinuities cover up:


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*Phase transitions in thermodynamics. These are only truly discontinuous in the $N \to \infty$ limit, which doesn't physically exist. For large but finite $N$, we can use statistical mechanics to get a perfectly continuous answer.

*Measurement in quantum mechanics. 'Copenhagen collapse' is not instantaneous, it's the result of interaction with an external system, which occurs in continuous time.

*Optical decays. Without QED, the best model is to just have atoms suddenly and randomly emit photons with some lifetime. With QED, we have a perfectly continuous time evolution (allowing for, e.g. Rabi oscillations).


At this point I'm having trouble thinking of any 'real' discontinuities. Are there any theories (that we believe to be fundamental) that predict a discontinuity in a physically observable quantity?

To address several comments: I am not looking for a discontinuity in time, as this is associated with infinite energy. I am not looking for experimental confirmation of a discontinuity in time, since that's impossible. 
I am asking if there is any measurable parameter in any of our currently most fundamental theories which changes discontinuously as a function of another measurable parameter, according to the theory itself. For example, if phase transitions actually existed, then phase as a function of temperature or pressure would work.
 A: Any process that causes a physical quantity to become truly discontinuous in space and/or time by definition takes place over an extremely (in fact, infinitely) short time or length scale.  From the usual uncertainty principles of quantum mechanics, these process would have huge energy or momentum, and would presumably result in both very strong quantum and gravitational effects.  Since we don't have a good theory of quantum gravity, there's really very little we can say with confidence about such extreme regimes.
But even if we do one day come up with a perfectly well-defined and self-consistent theory that reconciles quantum field theory with general relativity and is completely continuous in every way, that still won't settle your question.  Such a theory can never be proven to be "the final theory," because there will always be the possibility that new experimental data will require it to be generalized.  The most likely place for this "new physics" would probably be at whatever energy scales are beyond our current experimental reach at the time.  So we'll probably always be the least confident of the physics at the very smallest of length or time scales.
A similar line of thought holds for the possibility of absolute discontinuities in energy or momentum: ruling out, say, really tiny discontinuities in energy would require knowing the energy to extremely high precision.  But by the energy-time uncertainty relation, establishing the energy to such high precision would require an extremely long time - and eventually the required time scale would become too long to be experimentally feasible.
So extremely long and extremely short time/length scales both present fundamental difficulties in different ways, and your question will probably never be answerable.
A: Although you've indicated elsewhere that you don't like this example, I'm posting it in case others like it more:
Balance a pencil on its point, at an angle $\theta$ ranging from $0$ (flat on the table) to $\pi/2$ (perfectly vertical).  Let $f(\theta)$ be the angle of the pencil when it reaches equilibrium.  Then for any $\theta<\pi/2$, we have $f(\theta)=0$, so $\lim_{\theta\rightarrow\pi/2}=0$.  But $f(\pi/2)=\pi/2$, so $f$ is not continuous.
Of course any unstable equilibrium gives rise to a similar example.
A: 
I am asking if there is any measurable parameter in any of our currently most fundamental theories which changes discontinuously as a function of another measurable parameter, according to the theory itself.

If you take Quantum Mechanics as the Copenhagen interpretation puts it, then every measurement causes a discontinuity of some kind. According to the interpretation, every measurement is said to instantaneously collapse the wave function – the wavefunction abruptly and discontinuously changes in time.
But since you are talking explicitly about a change in a  “measurable parameter”, the wavefunction itself doesn't count, as it's not measurable in a physical sense. However, many quantities that are derived from it are measurable.
For example, let's take the expectation value of spin along the $z$ axis. If we take a beam of spin-½ particles that are all oriented up, the expectation value would be precisely $$ \left< \hat S_z \right> = +\frac{1}{2} \: .$$
When we take this beam and measure its spin along the $x$ axis, the state of those particles will immediately become an uncorrelated susperposition of $\left|z+\right>$ and $\left|z-\right>$ and the new expectation value would be $$ \left< \hat S_z \right> = 0 \: .$$
Now, I'm not saying that the Copenhagen interpretation provides the deepest insight into Quantum Mechanics, probably not, but it is the standard view and it's not obvious whether the discontinuities go away in other interpretations, or they just “hide elsewhere”.
A: The nearest discontinuous "instantaneous" process I know of is described in Interpreting attoclock measurements of tunnelling times where...

Resolving in time the dynamics of light absorption by atoms and
  molecules, and the electronic rearrangement this induces, is among the
  most challenging goals of attosecond spectroscopy. The attoclock is an
  elegant approach to this problem, which encodes ionization times in
  the strong-field regime. However, the accurate reconstruction of these
  times from experimental data presents a formidable theoretical task.
  Here, we solve this problem by combining analytical theory with ab
  initio numerical simulations. We apply our theory to numerical
  attoclock experiments on the hydrogen atom to extract ionization time
  delays and analyse their nature. Strong-field ionization is often
  viewed as optical tunnelling through the barrier created by the field
  and the core potential. We show that, in the hydrogen atom, optical
  tunnelling is instantaneous.

