1. ## Cont. Func.

Determine whether there exists a function that is continuous on all of $\displaystyle \mathbb{R}$ where it has range $\displaystyle f(\mathbb{R})$ that's equal to $\displaystyle \mathbb{Q}$

2. There isn't such a function, because if f is continuous than $\displaystyle f(\mathbf{R})$ must be an interval and $\displaystyle \mathbf{Q}$ is not an interval.

3. Originally Posted by red_dog
There isn't such a function, because if f is continuous than $\displaystyle f(\mathbf{R})$ must be an interval and $\displaystyle \mathbf{Q}$ is not an interval.
? let $\displaystyle f(x) = c$ which continuous on R. is $\displaystyle f(\mathbf R)$ an interval?

4. Originally Posted by kalagota
? let $\displaystyle f(x) = c$ which continuous on R. is $\displaystyle f(\mathbf R)$ an interval?
Continous functions map intervals into intervals or single points. In either case what reddog said is absolutely true.

EDIT: It would be easier to use the fact that cotinous functions map connected sets to connected sets, R is connected, Q is not.