MathJoy

It is illustrated by the figure that the set of the rational numbers is not a continuum, there are holes in it, one question is how many points are inside the hole? From a geometrical perspective, I think treating real numbers as shorthands of Dedekind cuts of the set of rational numbers cannot convincingly answer the question - how many points are inside the hole revealed by Dedekind cut not produced by a rational number like \(\begin{matrix} L & \ = \{ x \mid x \in \mathbb{Q},x \leq 0\} \cup \left\{ x \mid x \in \mathbb{Q},x > 0,x^{2} < 2 \right\} \\ U & \ = \mathbb{Q} - L = \left\{ x \mid x \in \mathbb{Q},x > 0,x^{2} > 2 \right\} \\ \end{matrix}\) . Every such Dedekind cut can be conceivably illustrated on the number axis, without a sound proof, to say such a cut corresponds with one and only one point on the number axis is far from trust , however, which is what exactly required by Cantor-Dedekind axiom 1 , 2 . Unless a convincing proof is given, otherwi...

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 (1) Give an option to allow seting the SUDO mode as a default setting (2) To learn form WinSCP to allow bookmarking some frequently used path as bookmarks, so users can access them conveniently. 

First, it is necessary to introduce the following definitions 1 , The function is said to be increasing at \(x_{0}\) if for all \(x\) -values in some interval about \(x_{0}\) it is true that when \(x_{0} < x\) then \(y_{0} < y\) , and when \(x_{0} > x\) then \(y_{0} > y\) . The function is said to be decreasing at \(x_{0}\) if for all \(x\) -val ues in some interval about \(x_{0}\) it is true that when \(x_{0} < x\) then \(y_{0} > y\) , and when \(x_{0} > x\) then \(y_{0} < y\) . Then the definition of “a function is non-decreasing at \(x_{0}\) ” is introduced by extend the definition of increasing above. The function is said to be non-decreasing at \(x_{0}\) if for all \(x\) -values in some interval about \(x_{0}\) it is true that when \(x_{0} < x\) then \(y_{0} \leq y\) , and when \(x_{0} > x\) then \(y_{0} \geq y\) . The intermediate value theorem states: If \(f\) is a continuous function on a closed interval \(\lbrack a,b\rbr...