if-E-f-f-R-R-continuous-function-with-f-x-Q-x-R-then-E-

Question Number 11067 by suci last updated on 10/Mar/17

i f E = {f ∣ f : R \to R c o n t i n u o u s f u n c t i o n w i t h f (x) \in Q, \forall x \in R},

t h e n E = \dots . ? ? ?

Commented by FilupS last updated on 10/Mar/17

Q = irrational numbers set

one example (i think) :

f (x) = \frac{2 x + 1}{2 x}

or

f (x) = \frac{2 x}{2 x + 1}

Commented by FilupS last updated on 11/Mar/17

in general

E = {f ∣ x \to \frac{a}{b}, a \land b \in Z, a ∤ b}

Commented by FilupS last updated on 11/Mar/17

Also, \forall x \in R : f (x) \in Q seems tricky .

generally : f (x) = \frac{a}{b} (where a ∤ b)

a, b \in Z

if x = π \Rightarrow f (π) = \frac{a}{b}

∴ k π = \frac{a}{b} \Rightarrow k \in R ∖ Q

∴ \frac{a}{b} \overset{?}{\notin} Q

thus I beleive f (x) \in Q

i am not sure

Commented by prakash jain last updated on 12/Mar/17

If f (x) is continuous and f (x) \in Q

\Rightarrow f (x) = c w h e r e c i s a c o n s t a n t \in Q

o r E i s s a m e a s Q