Help with understanding why limit implies uniqueness

In summary, the conversation discusses the uniqueness of solutions in the context of ODEs. The main topic of concern is the logical fallacy of "denying the antecedent." The conversation also mentions the possibility of other arguments justifying the book's conclusion.
  • #1
MathStudent999
1
0
TL;DR Summary
I was hoping to find more clarification on uniqueness results for autonomous ODEs
I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose
$$\begin{cases}
f \in \mathcal{C}(\mathbb{R}) \\
\dot{x} = f(x) \\
x(0) = 0 \\
f(0) = 0 \\
\end{cases}.
$$

Then,
$$
\lim_{\varepsilon \searrow 0}\left|\int^\varepsilon_0 {dy \over f(y)}\right| = \infty
$$
implies solutions are unique. Since
$$
\lim_{\varepsilon \searrow 0}\left|\int^\varepsilon_0 {dy \over f(y)}\right|< \infty
$$
allows us to invert to get a solution(more clarification on this) other than 0. So, am I seeing it right that this is just a contrapostive to get uniqueness.
 
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  • #2
If the book presents it as you describe, then your concerns are well-founded, as it makes the logical fallacy of "denying the antecedent" (I can't post a link to the wiki page on this locked-down computer), which is erroneously concluding ##\neg P\to \neg Q## from ##P\to Q##.

In this case, ##P## is the inequality in the OP and ##Q## is the claim "there exists more than one solution".

From ##P\to Q## one can conclude ##\neg Q\to\neg P## but one cannot conclude ##\neg P\to\neg Q##. eg consider where ##P## is "Beryl was born in Bulgaria" and ##Q## is "Beryl was born in Europe".
Beryl may have been born in Poland.

I expect there are other arguments that can justify the book's conclusion, but the author didn't notice the logical fallacy of the above, and hence omitted to state them.
 

Related to Help with understanding why limit implies uniqueness

1. What is a limit in mathematics?

A limit in mathematics is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a specific value.

2. How does a limit imply uniqueness?

A limit implies uniqueness because it shows that as the input approaches a certain value, the output of the function also approaches a specific value. This means that there can only be one value that the function approaches at that particular input, making it unique.

3. Can a limit have multiple values?

No, a limit cannot have multiple values. If a function has a limit at a certain input, it means that the output of the function approaches a specific value as the input gets closer to that value. Therefore, there can only be one value that the function approaches, making it unique.

4. How is the uniqueness of a limit proven?

The uniqueness of a limit is proven using the epsilon-delta definition. This involves showing that for any small positive value of epsilon, there exists a corresponding positive value of delta such that the output of the function is within epsilon distance of the limit as the input approaches the specific value within delta distance.

5. Why is understanding limit and uniqueness important in mathematics?

Understanding limit and uniqueness is important in mathematics because it is a fundamental concept that is used in many areas of mathematics, including calculus, analysis, and differential equations. It allows us to determine the behavior of a function and make predictions about its values, which is crucial in solving real-world problems and developing new mathematical theories.

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