  Let us now distinguish various stages in the chronogical order of the life of the film. In stage S1 the film is young; it has just been placed in the camera and is ready to be exposed. It is then exposed to the object that comes out of the time machine. (That object in fact is a later stage of the film itself). By the time we come to stage S2 of the life of the film, it has been developed and is about to enter the time machine. Stage S3 occurs just after it exits the time machine and just before it is photographed. Stage S4 occurs after it has been photographed and before it starts fading away. Let us assume that the film starts out in stage S1 in some uniform shade of gray, and that the only significant change in the shade of gray of the film occurs between stages S1 and S2. During that period it acquires a shade of gray that depends on the shade of gray of the object that was photographed. I.e., the shade of gray that the film acquires at stage S2 depends on the shade of gray it has at stage S3. The influence of the shade of gray of the film at stage S3, on the shade of gray of the film at stage S2, can be represented as a mapping, or function, from the real numbers between 0 and 1 (inclusive), to the real numbers between 0 and 1 (inclusive). Let us suppose that the process of photography is such that if one imagines varying the shade of gray of an object in a smooth, continuous manner then the shade of gray of the developed picture of that object will also vary in a smooth, continuous manner. This implies that the function in question will be a continuous function. Now any continuous function from the real numbers between 0 and 1 (inclusive) to the real numbers between 0 and 1 (inclusive) must map at least one number to itself. One can quickly convince oneself of this by graphing such functions. For one will quickly see that any continuous function f from [0,1] to [0,1] must intersect the line x=y somewhere, and thus there must be at least one point x such that f(x)=x. Such points are called fixed points of the function. Now let us think about what such a fixed point represents. It represents a shade of gray such that, when photographed, it will produce a developed film with exactly that same shade of gray. The existence of such a fixed point implies a solution to the apparent paradox.