f-0-1-R-x-a-0-a-1-a-2-a-3-f-x-i-0-a-i-is-f-x-continuous-in-all-x-0-1-f-0-9999-f-1-1-

Question Number 2393 by 123456 last updated on 19/Nov/15

f : [0, 1] \to R

x = a_{0}, a_{1} a_{2} a_{3} \dots

f (x) = \underset{i = 0}{\sum^{+ \infty}} a_{i}

is f (x) continuous in all x \in [0, 1]

f (0, 9999 \dots) = ? ?

f (1) = 1

Commented by prakash jain last updated on 19/Nov/15

f (x) is not continous is limit does not

exists for irrational x and also for rational

x = p / q where q \neq 2^{n} 5^{m}

Commented by prakash jain last updated on 19/Nov/15

I {don}^{'} t think there is any real number like

0.99999 \dots . (infinite times) . It is same as 1 .

Commented by prakash jain last updated on 19/Nov/15

What i mean is 0.999 . . is a recurring decimal

and hence rational . The rational number

representing this number is 1 .

This is different from other rationals

like 1 / 3 . which can be represented both

in recurring decimal and fraction notation .

So for example

0.49999999999 \dots (infinite times) = 1 / 2

Answered by Filup last updated on 19/Nov/15

0 . \bar{9} is e x a c t l y equal to 1 .

Because of the infinite number of 9^{'} s

being appended, this makes it entirely

equal .

Think about it like this :

If 1 \neq 0 . \bar{9}, t h e n :

0 . \bar{9} = 0.999 \dots 999000 \bar{0} \dots which is

contradictory against the definition

of recurring decimals . A L L recurring

decimals are rational, t h u s :

f (0 . \bar{9}) = f (1) i f f 0 . \bar{9} is appended infinitly!

Commented by prakash jain last updated on 19/Nov/15

Correct .

0.4 \bar{9} = 1 / 2

0 . \bar{9} = 1 .

1 / 2 = . 5

Commented by 123456 last updated on 19/Nov/15

0, 4 \bar{9} = \frac{1}{2}

0, 4 + x = \frac{1}{2}

\frac{2}{5} + x = \frac{1}{2}

x = \frac{1}{2} - \frac{2}{5} = \frac{5 - 4}{10} = \frac{1}{10} = 0, 1

Commented by prakash jain last updated on 20/Nov/15

0.1111 \times 9 = . 9999

\frac{1}{9} = 0.1111 \dots . .

\frac{1}{9} \times 9 = 0.99999 \dots . .

But 0 . \bar{9} and 1 are same numbers .

f (x) = \underset{i = 0}{\sum^{\infty}} a_{i} if x = a_{0} . a_{1} a_{2} \dots .

f (0 . \bar{9}) = ?

Commented by RasheedAhmad last updated on 19/Nov/15

\underline{O n e d i f f e r e n c e b e t w e e n 0.9}

\underline{a n d 0.3}

0 . \overset{―}{3} c a n b e a c h i e v e d b y a c t u a l

d i v i s i o n p r o c e s s .

1 \div 3 = 0 . \overset{―}{3}

B u t 0 . \overset{―}{9} c a n n o t b e a c h i e v e d b y

a c t u a l p r o c e s s o f d i v i s i o n

o f a n y t w o i n t e g e r s .

0 . \overset{―}{9} = 1 = \frac{1}{1}

1 \div 1 {d o e s n}^{'} t y i e l d 0.9999 \dots

i t i s s i m p l y 1

0.4 \overset{―}{9} i s a l s o l i k e 0 . \overset{―}{9}

I t c a n n o t b e a c h i e v e d b y a c t u a l

p r o c e s s o f d i v i s i o n o f a n y t w o

i n t e g e r s .

1 \div 2 {d o e s n}^{'} t y i e l d 0.49999 \dots

i t i s s i m p l y 0.5

Commented by 123456 last updated on 19/Nov/15

\frac{a}{9} = 0, \bar{a} a \in {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ?}

\frac{9}{9} = 0, 9999 \dots

9 ∣ 9

0 1

9 ∣ 9

90 ⌊ 0, 99 \dots

81

90

81

⋮

4 ∣ 4

40 ⌊ 0, 9 \dots

36

40

⋮

Commented by Rasheed Soomro last updated on 20/Nov/15

^{'} V N i c e!^{'} f o r y o u r d e e p a p p r o a c h!

Y o u h a v e s u c c e s s f u l l y s h o w e d / p r o v e d i n v a r i o u s w a y s t h a t

0 . \overset{―}{9} = 1

A c t u a l l y I {d o n}^{'} t d e n y t h e a b o v e . I o n l y p o i n t e d o u t

t h e d i f f e r e n c e b e t w e e n 0 . \overset{―}{3} a n d 0 . \overset{―}{9}

0 . \overset{―}{3} i s t h e r e s u l t o f o r d i n a r y d i v i s i o n o f t w o i n t e g e r s

1 \div 3 = 0.3333 \dots w h e r e a s 0 . \overset{―}{9} i s n o t .

Y o u h a v e a l s o t r i e d t o s h o w t h a t 0 . \overset{―}{9} c a n b e a c h i e v e d

b y o r d i n a r y d i v i s i o n (9 \div 9) b u t i n t h i s c o n n e c t i o n y o u

{h a v e n}^{'} t f o l l o w e d s o m e r u l e s o f o r d i n a r y d i v i s i o n .

\underset{- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -}{S o I t h i n k 0.9, 0.49 e t c {a r e n}^{'} t a c h i e v a b l e b y o r d i n a r y}

\underset{- - - - - - - - - - - - - - - - - - -}{d i v i s i o n o f t w o i n t e g e r s .}

Commented by 123456 last updated on 19/Nov/15

0, 9999 \dots

= 9 / 10 + 9 / 10^{2} + \dots

= \frac{9 / 10}{1 - 1 / 10} = \frac{9 / 10}{9 / 10} = 1 (infinite pg)

or

s = 0, 999 \dots .

10 s = 9, 999 \dots .

9 s = 9

s = 1

ps :

1 = 0, 11111111 \dots ._{2}

since

\underset{i = 1}{\sum^{+ \infty}} \frac{1}{2^{i}} = \frac{1 / 2}{1 - 1 / 2} = 1

if f doenst is continuous them (dont sure)

\lim_{n \to + \infty} f (x_{n}) \neq f (\lim_{n \to + \infty} x_{n}) in general

them this have something about

f (0, 99 \dots) and f (1) o r

⌊ 0, 999 \dots ⌋ and ⌊ 1 ⌋

or i think it have : v

Commented by Rasheed Soomro last updated on 20/Nov/15

0 . \overset{―}{9} = 1 \Rightarrow f (0 . \overset{―}{9}) = f (1) = 1

O n t h e o t h e r h a n d f (0 . \overset{―}{9}) = 9 . \infty = \infty

T h a t m e a n s t o a v o i d c o n t a d i c t i o n w e s h o u l d

d e n y t h e s e p a r t e e x i s t e n c e o f 0 . \overset{―}{9} a n d t h i n k^{'} i t i s

n o t h i n g b u t 1^{'} ?

S i m i l a r l y f (0.4 \overset{―}{9}) = f (0.5) = 5 ?

Commented by 123456 last updated on 20/Nov/15

one day in some random math i found

about ⌊ 0, 99 \dots ⌋ = 0 and ⌊ 1 ⌋ = 1 but 0, 9 . . = 1

the answer them give are about continuity

of ⌊ x ⌋, i will search the thing later .

Commented by Rasheed Soomro last updated on 21/Nov/15

I^{'} l l w a i t f o r u p d a t e!