I just made a file RealNumber for real number, hope that it can simplify the
bascic problem about real number and provide solutions regarding problems of
infinity, 0.999...,etc. which is explained shortly and indirectly.

--------------------
The purpose this text is for establishing the bases for computational algorithm.
This file https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
may be updated anytime.

+-------------+
| Real Number |
+-------------+

n-ary Floating Point Number::= Number represented by a string of digits, the
string may contain a point and/or plus/minus sign.... Two n-ary floating
point number x,y are equal iff their normalized form are identical.

Real Nunmber(ℝ)::= {x| x is represented by n-ary floating point number. The
string of digits of x may be infinitely long }

Note: This definition implies that repeating decimals are irrational number.

Real number is just this simple. The limit theory is a methodology for finding
derivative, nothing to do with what the real number is (otherwise, a definition
like the above must be defined in advance. Otherwise, latter dedution will be
difficult not to contain circular-reasoning.

+--------------------------------------+
| Restoring Interpretation of Calculus |
+--------------------------------------+
Calculus is basically from the area problem of a function: Let F compute the
the area of f. From the meaning of area, we can have:

(F(x+h)-F(x)) ≒ (f(x+h)+f(x))*(h/2) // h is a sufficiently small offset
<=> (F(x+h)-F(x))/h ≒ (f(x+h)+f(x))/2

Expected property of F: (1)Error |lhs-rhs| strictly decreases with a tiny offset
h (2)When h=0, lhs=rhs.
Because the h in the lhs cannot be 0, the basic problem of calculus is
finding such a F that satisfies the expected porperty above.

F-expected property(1) can be relaxed, because such a F is still a
acknowledged area function. So, the interpretation above can be specified as:

Lim(h->0) (F(x+h)-F(x))/h = f(x)

Note: (1)The meaning of Lim is the same as the lim in text books but the h
must be explicitly 0 in the final processed result of (F(x+h)-F(x))/h.
Otherwise, the logical conclusion is always 'approximation'.
(2)The above expression is treated as an identity that can be derived,
not a definition (3)Nothing about infinity or infinitesmal is mentioned.

Continuous (function)::= Function f is continuous at a iff f(a) can be
approached from nearby-value f(a+h) as described by ε-δ method..

Note1: Borrowing from the limit theory, the idea can be expressed as
lim(h->0) f(x+h)= f(x)
Note2: For complex problems, most iterative algorithms depend on continuous
property to find the solution.
Note3: P≠NP should have been proved
https://sourceforge.net/projects/cscall/files/MisFiles/PNP-proof.txt/download
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