The seemingly continuous graphs of the rational hyperbolic functions: A=B/(1-B) and B=A/(1+A) when multiplied giving a simplified product-difference equation AB=A-B as their equivalent product-difference function suggests the existence of rational quantum gateways. Conversely, if these gateways do not exist then the separate graphs of this equivalent product-difference function are consequently at the least piecewise discontinuous at every infinitesimal neighborhood of any rational point given by the Cartesian coordinate (A, B). For example the point (1,½) satisfies the rational product-difference function. Some solutions between zero and unity values for A are: (0,0), (1/4,1/5), (1/3,1/4), (½,1/3), (2/3,2/5). (3/4,3/7), and of course the point at (1,½). For every given rational A value approaching zero, the corresponding value for B always has a denominator 1 more than the denominator for A. For example if A=1/1000 then B is 1/1001 or for A=1/100000 then B=1/100001 and so on. This is similar to the limit theorem of differential calculus and can be used to establish the existence of a derivative and functional continuity for AB=A-B. Since the two branches of the rational hyperbolas never intersect together they can be represented by the Hopf link topology for its three dimensional algebraic curves.
　
For all rational values of A greater than 1, the corresponding values of B approach unity as the values of A approach infinity. The slope of the rational function at A equals infinity is zero and the slope at A equals zero is unity which can be mapped into a one to one correspondence with the light cone of special relativity such that A=0 corresponds to a speed of light for zero masses of elementary particles and A=infinity corresponds to zero speed of infinite masses of elementary particles. Consequently, it can be shown that the area between the curve ()=/(1+) and the line g()=B/A for every point (A,B) along the rational functions and =A approaches zero. This implies that only in the infinitesimal neighborhood of A=0 does the arc length of the rational function approach that of a true vector which can be used to define the fixed length or magnitude of primary forces for the quantum theory of the spacetime continuum.