Is the Function Continuous at Any Point?

[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 ?