I-dont-know-how-to-show-the-functions-of-this-type-if-it-is-continuous-or-not-in-topological-space-I-know-the-method-is-to-use-one-of-two-properties-the-inverse-image-of-eah-open-is-open-and-the-i

Question Number 79081 by arkanmath7@gmail.com last updated on 22/Jan/20

I d o n t k n o w h o w t o s h o w t h e f u n c t i o n s o f

t h i s t y p e i f i t i s c o n t i n u o u s o r n o t i n t o p o l o g i c a l s p a c e .

I k n o w t h e m e t h o d i s t o u s e o n e o f t w o p r o p e r t i e s :

t h e i n v e r s e i m a g e o f e a h o p e n i s o p e n,

a n d t h e i n v e r s e i m a g e o f e a c h c l o s e d i s c l o s e d

b u t i d o n t k n o w h o w t o u s e t h e m .

i w i l l b e t h a k e d i f s o m e o n e h e l p e d m e w i t h t h e s t e p s

i f f : (R, U) - > (R, U) d e f i n e d b y

f (x) = {\begin{cases} \frac{1}{3} x + 1, x > 3 \\ \frac{1}{2} (x + 5), x ≦ 3 \end{cases}

I s f c o n t i n u o u s ?

Answered by mind is power last updated on 22/Jan/20

S i n c e I R i s m e t r i c a l w e c a n u s e s e q u e n c e s e

m e t r i c s p a c e \Rightarrow s e p a r a b a l \Rightarrow u n i c i t e o f l i m i t e s

U_{n} = 3 - \frac{1}{n} \to 3, V_{n} = 3 + \frac{1}{n} \to 3, w i t h e n \in N^{*}

U_{n} ⩽ 3, V_{n} > 3

f (U_{n}) = \frac{1}{2} (3 - \frac{1}{n} + 5) = 4 - \frac{1}{2 n} \to 4

f (V_{n}) = \frac{1}{3} (3 + \frac{1}{n}) + 1 = 2 + \frac{1}{3 n} \to 4

You can't use 'macro parameter character #' in math mode

Commented by arkanmath7@gmail.com last updated on 22/Jan/20

Commented by arkanmath7@gmail.com last updated on 22/Jan/20

I w r o t e i s i t c o n t ? b u t t h e q u s t n i n t e s t s a y s s h o w i t i s c o n t .