Real Analysis-Continuity

[popitar]

Homework Statement

Let f:R->R be defined by f(x)=x^2 for x in Q and x+2 if x in R\Q. Find all points (if any) where f is continuous.

Homework Equations

The Attempt at a Solution

 

[╔(σ_σ)╝]

f can only be continuous at 2.

Try showing this by using rational and irrational sequences approaching any point c.

 

[popitar]

how about -1?
but how to show it??

 

[╔(σ_σ)╝]

Use sequences of rationals number converging to an irrational number to show the function is not contituous at points other than -1,2.

 

[popitar]

Could you please help me with it? Please show me how this function is continuous at -1. Any help is appreciated..

 

[jbunniii]

$f(-1) = 1 = (-1)^2 = (-1) + 2$

All you have to do is show that both $x^2$ and $x + 2$ are arbitrarily close to 1 when $x$ is sufficiently close to -1. This is quite simple because both of those functions are continuous at -1.

Formally, given $\epsilon > 0$, you need to find a $\delta > 0$ such that when $|x - 1| < \delta$, both $x^2$ and $x + 2$ are within $\epsilon$ of 1. It's pretty easy to find such a $\delta$.

 

[╔(σ_σ)╝]

I wrote up a proof but it got eaten up by my browser.
Do you know how to proof continuity using sequences ?

 

[popitar]

Thank you all!

 

[╔(σ_σ)╝]

Did you figure out how to do the problem ?