Of course there are points within a continuum, but it is the nature (and the definition) of any continuum that adjacent points are not significantly different. Thus any chosen beginning becomes arbitrary.
Nevertheless, at one point, there's a white patch then there's a letter.
With photon generation, you chose an example where quantum scales may be significant for one photon of each dot of the letter (on a CRT, electron excites phosphur atom which emits a photon without apparent smooth transitions). However, for the entire letter that you see, there is a delay between each dot on the grid, which are not lit simultaneously. These photons leave the screen phosphors at different times, travel a distance from the phosphors to your eyes, through your cornea, aqueous humor, lens, vitreous humor, plasma membrane of a photosensitive retinal cell, and excite a molecule of rhodopsin (another quantum event). Then, the 11-cis-retinal moiety of rhodopsin changes shape to form 11-trans-retinal, a series of chemical reactions produces metarhodopsin from rhodopsin, metarhodopsin activates cyclic-GMPases which, through another series of chemical reactions that lead to ion transfer, changes the transmembrane potential of the cell. Then, when enough photons excite enough rhodopsin molecules, the membrane potential changes sufficiently to stimulate electrical signal propagation to an optic gangion cell, then through the optic nerve, optic chiasm, opic tract, optic radiations, to the visual cortex of the occipital lobe. These propagating signals can also be broken down into a series of chemical events. From there signals propagate to other areas of your brain that result eventually in perception of the letter on your computer screen. I say eventually, because a single photon, especially in a lit room, will not be sufficient for you to "see" even one dot of that letter, let alone the entire letter. It will take several retraces of your CRT to generate enough photons for you to perceive the letter.
Of course, I've left out many steps in this process. But where among this continuum would you place a nonarbitrary dividing line to call the "beginning" of that letter? All time points in the gradual transition from 'no letter' to 'letter' are essential, but no adjacent points are different enough to place a nonarbitrary dividing line. The letter does not just poof into existence.
Nature transitions everything too smoothly (above quantum scales) for specific dividing points to be nonarbitrary.