Advanced-Calculus-Let-f-R-Q-is-a-continuous-function-prove-that-f-is-a-constant-function-m-n-Adopted-from-mathematical-an

Question Number 165339 by mnjuly1970 last updated on 30/Jan/22

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

L e t, f : R \to Q i s a c o n t i n u o u s

f u n c t i o n . p r o v e t h a t^{″} f^{″} i s a

c o n s t a n t f u n c t i o n . m . n

* A d o p t e d f r o m m a t h e m a t i c a l a n a l y s i s b o o k *

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Answered by mahdipoor last updated on 30/Jan/22

p r o o f b y c o s t r a d i c t i o n :

i f f o r \forall a < b, f (a) \neq f (b)

\exists k \in Q^{^{'}} t h a t b e t w e e n f (a) a n d f (b)

\Rightarrow f i s c o n t i n u o u s f u n c t i o n,

m e d i u m v a l u e t h e o r e m :

\exists c \in [a, b] t h a t f (c) = k \in Q^{^{'}}

b u t f : R \to Q (a n d c a n n o t f (c) \in Q^{^{'}})

t h e r e f o r, f o r \forall a < b, f (a) = f (b);

t h i s m e a n s t h a t f i s c o n s t a n t f u n c t i o n .

Commented by mnjuly1970 last updated on 30/Jan/22

b r a v o s i r \dots z e n d e b a s h i d

Commented by mahdipoor last updated on 30/Jan/22

p a y a n d e b a s h i d :)