How can we show the existance of a ε?

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In summary, the conversation discusses a proof that shows there exists a positive value for epsilon such that the difference between two continuous functions on [0,1] is always greater than or equal to epsilon when the two functions are not equal for any x on the interval. This is achieved by showing that the absolute value of the function is always greater than zero on the interval, and then setting epsilon to be the minimum of the absolute value of the function.
  • #1
mathmari
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MHB
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Hey! :eek:

Suppose that $h_1,h_2:[0,1]\rightarrow \mathbb{R}$ be continuous functions.

I want to show that there is a $\epsilon>0$ such that $|h_1(x)-h_2(x)|\geq \epsilon$ for each $x\in [0,1]$, given that $h_1(x)\neq h_2(x)$ for each $x\in [0,1]$. I have done the following:

We know that $h_1(x)\neq h_2(x)$ for each $x\in [0,1]$, that means that $f(x):=h_1(x)-h_2(x)$ is either $>0$ or $<0$ but never $=0$.

Therefore, we have that $|f(x)|>0, \forall x\in [0,1]$. That means that there is a $\epsilon>0$ such that $|f(x)|\geq \epsilon, \forall x\in [0,1]$. Is this correct? (Wondering)
 
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  • #2
Yes, it's correct. However, you may need to justify the last step. Since $|f|$ is continuous on a closed and bounded interval, it has a minimum at some point $a\in [0,1]$. Setting $\epsilon = |f(a)|$ does the job.
 
  • #3
Euge said:
Yes, it's correct. However, you may need to justify the last step. Since $|f|$ is continuous on a closed and bounded interval, it has a minimum at some point $a\in [0,1]$. Setting $\epsilon = |f(a)|$ does the job.

I see... Thank you very much! (Yes)
 

FAQ: How can we show the existance of a ε?

What is an ε?

An ε, or epsilon, is a symbol used in mathematics to represent a very small quantity, often approaching zero.

Why is it important to show the existence of an ε?

Showing the existence of an ε is important in many mathematical proofs, as it can help demonstrate the convergence or limit of a sequence or function.

How can we show the existence of an ε?

One way to show the existence of an ε is through a proof by contradiction, where we assume that the ε does not exist and then arrive at a contradiction. Another method is to use the definition of a limit, where we show that for any given small positive number, there exists a corresponding value in the sequence or function that is within that range.

What types of mathematical problems use the concept of an ε?

The concept of an ε is commonly used in calculus, real analysis, and other areas of mathematics where the limit or convergence of a sequence or function needs to be proven.

Can an ε be negative?

No, an ε is always a positive number. It represents a distance or interval, and negative distances do not exist.

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