1. ## determine f(sqrt(2))?

Suppose f:R->R is continuous and f(r)=$\displaystyle r^2$ for each rational number r. Determine f($\displaystyle \sqrt{2}$) and justify your conclusions.

I have no clue

2. Originally Posted by tn11631
Suppose f:R->R is continuous and f(r)=$\displaystyle r^2$ for each rational number r. Determine f($\displaystyle \sqrt{2}$) and justify your conclusions
If $\displaystyle f:\mathbb{R}\to\mathbb{R}$ is continuous then if $\displaystyle (x_n)\to x_0$ then it is the case that $\displaystyle f\left(x_n\right)\to f\left(x_0\right) .$

3. Originally Posted by Plato
If $\displaystyle f:\mathbb{R}\to\mathbb{R}$ is continuous then if $\displaystyle (x_n)\to x_0$ then it is the case that $\displaystyle f\left(x_n\right)\to f\left(x_0\right) .$
so wait does that just mean it would be 2? And would we have to show it by showing that the limit of lim_r->$\displaystyle \sqrt{2}$ f(r)=2 which is equal to f($\displaystyle \sqrt{2}$)..Or can we say that since it is continuous we know that lim as r->$\displaystyle \sqrt{2}$ of f(r)=f($\displaystyle \sqrt{2}$)? which is why we know its 2? I hope its not getting confusing.