Provide an example or prove it wrong

[Quinzio]

Homework Statement

A continuous function g: Q --> R such that g(0)=0 and g(1)=1, but there does not exist any x in Q such that g(x)=1/2

Homework Equations

Number sets, basic number theory

The Attempt at a Solution

The function could be $$f(x) = x^2$$

since

$$f(0) = 0$$

$$f(1) = 1$$

$$f(x) = 1/2$$

here it looks like $$x$$ must be an irrational. One could refer to the demonstration that the diagonal of a square is an irrational.

Is it enough as a prove ?

 

[Mark44]

Homework Statement

A continuous function g: Q --> R such that g(0)=0 and g(1)=1, but there does not exist any x in Q such that g(x)=1/2

Homework Equations

Number sets, basic number theory

The Attempt at a Solution

The function could be $$f(x) = x^2$$

since

$$f(0) = 0$$

$$f(1) = 1$$

$$f(x) = 1/2$$

here it looks like $$x$$ must be an irrational. One could refer to the demonstration that the diagonal of a square is an irrational.

Is it enough as a prove ?

Going by the title of your thread, you need to do one of two things:
1) Provide an example for which the conditions are met.
2) Prove that no such example exists.

f(x) = x² is an example that works, so you don't need to prove anything.

 

[Quinzio]

Going by the title of your thread, you need to do one of two things:
1) Provide an example for which the conditions are met.
2) Prove that no such example exists.

f(x) = x² is an example that works, so you don't need to prove anything.

Ok, I see.

Thanks for your contributions.