# Thread: f(x) continuous at c

1. ## f(x) continuous at c

Prove that if c is irrational f(x) is continuous at c.

I would appreciate any type of hints or just where to start.

2. ## Re: f(x) continuous at c

think it would help if you posted what f(x) is ... ?

3. ## Re: f(x) continuous at c

Originally Posted by cloeannx3
Prove that if c is irrational f(x) is continuous at c.
I would appreciate any type of hints or just where to start.

You told us absolutely nothing about $\displaystyle f$!

4. ## Re: f(x) continuous at c

I'm so sorry. I didn't even realize that I left that part out. (It's been a long week)

f(x) is defined on the interval [0,1] to be 0 if x xis irrational and to be 1/n if x is rational and if x can be written fully reduced as x =m/n

5. ## Re: f(x) continuous at c

suppose ε > 0 is given. we need to find some δ > 0 so that |x-c| < δ means |f(x)| < ε.

well pick a natural number n such that 1/n < ε. then there are only finitely many points (all of them rational) in the interval [0,1] with f(x) ≥ 1/n (let's call them x1,...,xk).

let δ = min(|x1 - c|, |x2 - c|,...,|xk - c|). does that work?

(for example, if ε = 0.4, we could use n = 3. the only points for which f(x) ≥ 1/3 are {0,1/3,1/2,2/3,1}, pick the one that c is closest to (for example if c = √2 - 1, we would pick δ = 4/3 - √2, since 1/3 is the closest point)).