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We can take y = 1/x as another example. Although the plot is infinitely differentiable, it lacks continuity.

one over x

The above plots also reveal discontinuity of real numbers as the curve approaches the y-axis. The continuity breaks down as it approaches point zero. Therefore, the field of real numbers is not inherently smooth or continuous (holomorphic). If we desire continuity, we have to incorporate the imaginary number and enter the concept of complex numbers into the equation. Therefore, we would rewrite the equation y = 1/x as y = 1/z (where z is a complex number, shown as z = x + ib, and b is any number).

The Peano curve also suggests the discrete texture of the observable properties of elements.

Peano curve


In 1890, Giuseppe Peano showed that a curved line can be constructed that completely fills a plane. The question is how a one-dimensional line with no thickness can fill up a two-dimensional plane. A logical answer is that a plane is not a smooth and continuous surface but made of discrete points with nothingness separating them. Therefore, a one-dimensional line representing an observable thing can bypass the nothingness and reach through all zero-dimensional points in the plane. Therefore we deduce,

AssertionC4: Pure real numbers have to be discrete, and continuity always breaks as it approaches point zero

Here I conclude that  as observable prameters of physical elements are divided down to their atoms, the continuity of such parameters break down and exhibit discreteness.

On the other hand, we can choose any other point in the domain and shift point zero to that point, using the Cauchy formula in the origin shifted form.

 n!/2pi∫ f(z)/(z-p)n+1dz = f(p),

And the nth-derivative expression would be     n!/2pi∫ f(z)/(z-p)n+1dz = f (n)(p),p),

Roger Penrose writes, “Thus complex smoothness implies analyticity (holomorphicity) at every point of the domain.”[1]. Taking every point of the domain as zero is called “blowing up the origin.”

This is important when we are defining the proposed singularity and its relation with objective elements of the universe. We build the model with the assumption that singularity underlies every unit of space-time.

Assertion C5: The mathematics of complex numbers also indicates that any point in the field can be considered point zero.

On the other hand, the complex number equation Z = R [cos a + i sin a] indicates that these numbers have a periodic nature. Therefore, they lose their real number value and hit zero twice in each period. We take the periodic nature and intermittent appearance and disappearance of real value of measurable the basis for our sixth assertion.

 Assertion C6: Measurable properties of the elements are discrete (atomic). This includes fundamental elements, such as matter, time, and space

periodic complex number


For example, if the real coordinate denotes the mass of a particle, somewhere in its endeavor the tangible mass gradually loses its value and disappears.

By the same token, if x axis represents units of space, because of the periodic nature of complex system, the space has to disappear and reappear along the way. This is the basis for our assumption that space and time are not a continuum but made of atoms of space and time.

Penrose Roger, The Road to Reality 
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