x-2-x-x-x-x-x-times-taking-derivate-2x-1-1-1-1-x-times-2x-x-x-0-2-1-where-the-problem-

Question Number 2851 by 123456 last updated on 28/Nov/15

x^{2} = x + x + x + \cdot \cdot \cdot + x (x times)

taking derivate

2 x = 1 + 1 + 1 + \cdot \cdot \cdot + 1 (x times)

2 x = x (x \neq 0)

2 = 1

where the problem ?

Commented by Yozzi last updated on 29/Nov/15

T h a t f i r s t l i n e a s s u m e s t h a t x i s

a w h o l e n u m b e r . I f x i s v a r i a b l e a n d r e a l

a n d w e l e t x b e i r r a t i o n a l (t h a t

i s n o t o f t h e f o r m \sqrt{r}, r \in Q)

x^{2} c a n n o t b e e x p r e s s e d a s a r a t i o n a l

n u m b e r o f t h e f o r m \frac{a}{b} s i n c e

x^{2} = \frac{1}{b} + \frac{1}{b} + \frac{1}{b} + \dots + \frac{1}{b} (a t i m e s)

i s a c o n t r a d i c t i o n .

A n o t h e r n o n - e x a m p l e c o u l d b e

t h a t i f x = \frac{1}{2} \Rightarrow x^{2} = \frac{1}{4} b u t w h a t i s

\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \dots + \frac{1}{2} (\frac{1}{2} t i m e s) ?

x m u s t b e a w h o l e n u m b e r . S o t h a t

y = x^{2} i s n o t c o n t i n u o u s o n a n y

s u b - i n t e r v a l o f t h e r e a l a x i s,

e x c e p t a t p o i n t s .

T h e d e r i v a t i v e o f t h e y = x^{2} d o e s

n o t e x i s t f o r x \notin Z s i n c e y = x^{2} i s

u n d e f i n e d a t s u c h x .

Commented by Yozzi last updated on 29/Nov/15

L e t f (x) = {\begin{cases} x^{2} x \in Z^{+} \\ 0 o t h e r w i s e \end{cases}

Commented by Rasheed Soomro last updated on 29/Nov/15

V e r y N i c e!