Prove Continuity of \sqrt{x} in R+

[PhysicsDude1]

Homework Statement

Prove that $\sqrt{x}$ is continuous in R⁺ by using the epsilon-delta definition.

Homework Equations

A function f from R to R is continuous at a point a $\in$ R if :

Given ε> 0 there exists δ > 0 such that if |a - x| < δ then |f(a) - f(x)| < ε

The Attempt at a Solution

So I have to take an ε which is greater than 0 and prove that there exists a δ such that if the absolute value of (a - x) is smaller than that delta then the absolute values of the function values of a and x are smaller than ε.

I know I have to pick an ε which is greater than 0 but how do I know what value to pick for ε? 1,2,...n?

 

[Mark44]

Homework Statement

Prove that $\sqrt{x}$ is continuous in R⁺ by using the epsilon-delta definition.

Homework Equations

A function f from R to R is continuous at a point a $\in$ R if :

Given ε> 0 there exists δ > 0 such that if |a - x| < δ then |f(a) - f(x)| < ε

The Attempt at a Solution

So I have to take an ε which is greater than 0 and prove that there exists a δ such that if the absolute value of (a - x) is smaller than that delta then the absolute values of the function values of a and x are smaller than ε.

I know I have to pick an ε which is greater than 0 but how do I know what value to pick for ε? 1,2,...n?

You don't get to pick the ε. The whole δ - ε thing should be viewed as a dialog between you (who are trying to prove that a certain limit exists) and an acquaintance who is skeptical of the process. The other person gives you an ε value, which by the way is usually small and close to zero. You respond by finding a number δ so that when x is within δ of a, then √x is within ε units of √a.

If your skeptical friend is not satisfied, he will say something like, "Well it works for that ε. How about if ε is smaller?" You respond by finding a different δ, and show your friend that when x is within δ of a, then √x is again within ε units of √a.

The process continues until your skeptical friend realizes that no matter how small an ε he gives you, you are able to come up with a δ that works, and the limit is established.

 

[PhysicsDude1]

You don't get to pick the ε. The whole δ - ε thing should be viewed as a dialog between you (who are trying to prove that a certain limit exists) and an acquaintance who is skeptical of the process. The other person gives you an ε value, which by the way is usually small and close to zero. You respond by finding a number δ so that when x is within δ of a, then √x is within ε units of √a.

If your skeptical friend is not satisfied, he will say something like, "Well it works for that ε. How about if ε is smaller?" You respond by finding a different δ, and show your friend that when x is within δ of a, then √x is again within ε units of √a.

The process continues until your skeptical friend realizes that no matter how small an ε he gives you, you are able to come up with a δ that works, and the limit is established.

Thank you. This was very helpful intuive-wise! But how do I do this formally? Do I write δ in terms of ε?
I'm sorry but I'm really stuck here.

 

[Mark44]

Thank you. This was very helpful intuive-wise! But how do I do this formally? Do I write δ in terms of ε?
I'm sorry but I'm really stuck here.

Yes. Also, you need to work in the actual function, not f(x).

Start with the inequality |√x - √a| < ε and work backwards to |x - a| < <some expression>. That <some expression> will be your delta.

 

[HallsofIvy]

At any point, a, you want to prove that given some $\epsilon> 0$, there exist $\delta> 0$ such that "if $|x- a|< \delta$ then $|f(x)- f(a)|< \epsilon$.

So start with what you want to get: $|f(x)- f(a)|= |\sqrt{x}- \sqrt{a}|< \epsilon$ and try to manipulate that to get "$|x- a|<$ some number".

I recommend you start by separating this into two cases: x> a and x< a. Square root is an increasing function so that if x> a then $\sqrt{x}> \sqrt{a}$ and if x< a then $\sqrt{x}< \sqrt{a}$ so you can eliminate the absolute values.

 

[PhysicsDude1]

Yes. Also, you need to work in the actual function, not f(x).

Start with the inequality |√x - √a| < ε and work backwards to |x - a| < <some expression>. That <some expression> will be your delta.

|√x - √a| < ε
⇔

Sorry, accidentally clicked on post. I'm working on it :p

 

[PhysicsDude1]

Ok, so I've, actually it was you guys, come up with this so far :

|√x - √a| < ε

⇔ |√x - √a| = |√x - √a| . $\frac{|\sqrt{x}+ \sqrt{a}|}{|\sqrt{x}+ \sqrt{a}|}$

⇔ |√x - √a| = $\frac{|x-a|}{|\sqrt{x}+ \sqrt{a}|}$

⇔ $\frac{|x-a|}{|\sqrt{x}+ \sqrt{a}|}$ < ε

⇔ |x-a| < ε . $|\sqrt{x}+ \sqrt{a}|$

Is this correct so far?

 

[PeroK]

You've got the key equality, which is to relate √x - √a to x - a.

|√x - √a| = |x−a|/|√x +√a|

I would do this first for a = 0. I.e. prove √x is continuous at 0.

Then prove it for a > 1. If a > 1, then, |√x - √a| is smaller than |x−a|. So, it should be easy to find δ. But, if a < 1, then |√x - √a| could be larger than |x−a|. That's the tricky bit.

So, if you want to take it step by step, you could prove it for a >= 1. And, then finally prove it for a < 1. This might be easier until you get used to ε-δ.

For a < 1, there's a trick you'll need that is used a lot in ε-δ. It's not easy to spot first time you come across it.