Proving that f(x) is Not Continuous for All Real Numbers c

[phyguy321]

the function f(x) = 1 if x is rational
f(x) = 0 if x is irrational is not continuous for all real numbers, c



the function f(x) = x if x is rational
f(x) = 0 if x is irrationa is continuous at x=0 and not continuous for all other real numbers c

the function f(x) = 1/q if x is rational and x = p/q in lowest terms
f(x) = 0 if x is irrational
is continuous at c if c is irrational and not continuous at c if c is rational

I'm terrible at proofs, please help!

 

[tiny-tim]

Hi phyguy321;!

(have an epsilon: ε and a delta: δ )

Let's start with:

the function f(x) = x if x is rational
f(x) = 0 if x is irrationa is continuous at x=0 and not continuous for all other real numbers c

Can you use an ε and δ method to prove that it is continuous at x = 0?

Have a go.

 

[phyguy321]

Hi phyguy321;!

(have an epsilon: ε and a delta: δ )

Let's start with:


Can you use an ε and δ method to prove that it is continuous at x = 0?

Have a go.

I've got the proof for when f(x) is continuous at x=0 but I'm not sure how to prove that its discontinuous at every other value using delta epsilon. I understand it has to do with the density property of real numbers and that its full of holes be cause between every real number there exists rational and irrational numbers.

 

[tiny-tim]

I've got the proof for when f(x) is continuous at x=0 but I'm not sure how to prove that its discontinuous at every other value using delta epsilon. I understand it has to do with the density property of real numbers and that its full of holes be cause between every real number there exists rational and irrational numbers.

Hi phyguy321!

Same method … start "if x ≠ 0, then for any ε < x, … "

 

[phyguy321]

but how do i say if |x-c|<delta then |f(x) - f(not zero)|< epsilon?

 

[tiny-tim]

but how do i say if |x-c|<delta then |f(x) - f(not zero)|< epsilon?

But it's not continuous, so there isn't a δ.

You should be trying to prove that, no matter how small δ is, the neighbourhood will contain values that further away than ε.