Is motion un-observable? and if so how can we know it exists? Let's say I show you two pictures of the same desk, the only difference between them being the position of a red pen. Our intuition tells us that the pen must have been in motion at some point after the first picture was taken and before the second picture was taken. It is conceivable though that the pen was never in motion and instead vanished and reappeared where it rests in the second picture. Moreover, no matter how many pictures we have, they will likely never be more than circumstantial evidence for the theory of motion. In other words, it doesn't seem possible that discrete observations, no matter how many of them, would be able to "capture" motion.
This idea leads to the following related questions:
Is motion un-observable? And if so, how can we know it really exists? Might using discrete observations to make conclusions about the existence of continuous phenomena be problematic (as any such line of reasoning would heavily rely on intuition as opposed to physical observations)? Is it possible that motion is just an unimaginably amazing (almost perfect) but incorrect theory of the natural world?
I am only somewhat familiar with physics, so if I am missing something obvious please pardon my ignorance. Also, feel free to be as technical or informal as you see fit.
 A: You must be using  the word motion in a very specific sense, because in the more general sense of the word motion is clearly observable. Motion is a change of position over time, and we can clearly observe it.
If you are asking whether it is possible that what we perceive or measure as motion is not continuous at all scales, but instead beyond some threshold of short distance and/or time becomes discontinuous, then the answer is that we don't know, as we cannot make observations of continuous movement over infinitesimally small distances and times. However, what we can clearly believe is that if motion is discontinuous at some scale then it approximates to being continuous at larger scales.
A: The continuity of motion is a theoretical assumption. As you have pointed no number of discrete observations can force us to deduce that motion is continuous. What makes this assumption plausible is that if we make observations at time $s$ and $t$ with $t>s$, we can always make another observation at the mid-point of the time interval.
Moreover, QM shows that change, in a sense is not continuous, because what evolves continuously and deterministically is the quantum wave and it doesn't have real and actual ontological weight. So nothing real changes continuously, only the indeterministic quantum wave.
That motion should be thought of as some combination of the continuous and the discrete and of potential and actual motion was first done by Aristotle as his resolution of the paradoxes of motion by Zeno. The usual solution by infinite series was thought by Aristotle to be 'adequate' but didn't get to the 'true' heart of the matter.
A: We build up evidence for motion by a myriad of observations. One such observation is by putting something in the way. For example, if something is swinging to and fro on the end of a string, then put your hand in the middle and the thing will bump into your hand.
You could even put the reasoning the other way around. Given that motion is the common state of affairs, how do we know that anything ever stays still?
A: Don't know if this is satisfying, but here we go. If I'm allowed to be very technical, consider the free particle propagator in one dimension,
$$K(t,x)\sim e^{-\frac{x^2}{2it}}$$
This roughly gives the probability we'll measure the particle to have traveled $x$ in two subsequent measurements separated by time $t$. If you look real close, the exponent is of the form $x^2/t$, so the distance scale is much shorter than any timescale for a given small value of $t$. If we were actually to measure position infinitely often, the particle wouldn't be able to move at all.
So in actuality, motion is an emergent phenomenon of the fact that we are actually NOT continuously looking at things and measuring their precise positions, but doing only weak measurements that allow for positions to change.
A: In a comment you write:
"If you were to take this idea one step further, you might then come to the realization that there is no information in any number of discrete observations of any properties of the natural world that would ever suggest the existence of a continuum or of continuous phenomena unless you already have a preconceived intuition of these concepts."
Indeed, the  motion is a preconceived concept because we see continuous motion around us.
In the hidden variables approach to quantum mechanics, the wavefunction corresponds to an actual physical wave. This wave is thought to evolve continuously, but the particles immersed in the hidden variables (which in a sense can be thought to constitute space) jump instantaneously (or non-locally) from one position to another, within the confines of the wavefunction.
This happens "continuously" and it looks like the particle moves with fuzziness under the wavefunction. If the wavefunction has a global velocity, the particle moves with this velocity too, but it "dances" discontinuously under the save guide of the wavefunction.
But how do we know the wavefunction evolves continuously? We don't. But assume. You could ask what would cause a discontinuous behavior of the wavefunction. Nothing springs up, at least not in my mind. Maybe a collapse is a non-continuous, non-unitary evolution, but this is different from a particle disappearing here and popping up there. The wavefunction doesn't show this behavior when it evolves, but the particle does, in the HV theory, that is.
