Continuity leading to closure

In summary, the conversation discusses the relationship between continuity and closed sets. It is proven that if a function f is continuous, then the graph of the function, denoted as G, is a closed subset of R^2. This is shown by using the concept of continuity preserving limits and the characterization of closed sets in metric spaces. The open ball B is also mentioned as a potential connection between limits and closed sets.
  • #1
jaci55555
29
0
The graph of a continuous funtions (R -> R) is the subset G:={(x, f(x) | x element of R} is a subset of R^2. Prove that if f is continuous, then G is closed in R^2 (with euclidean metric).


I know that continuity preserves limits, so xn -> x in X means f(xn-> y in Y.
and for all A element of R^2 - G there exists r > 0 st B(a, r) subset of R^2 - G.I know that if R^2 - G is open then G is closed.


What is the connection between limits and the open ball B? I think that might be the way...
 
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  • #2
the following characterization might be useful:

A subset F of a metric space X is closed iff for all sequences (x_n) in F which converges to a certain x in X, it holds that x is in F.
 

1. What does "continuity leading to closure" mean in scientific terms?

In scientific terms, "continuity leading to closure" refers to the idea that processes or systems should remain consistent and connected in order to reach a desired endpoint or final state.

2. Why is continuity important in scientific research?

Continuity is important in scientific research because it allows for accurate and reliable results. By maintaining consistency and connection throughout a study, researchers can better understand cause and effect relationships and make valid conclusions.

3. How does continuity contribute to the scientific method?

Continuity is a key component of the scientific method as it helps to ensure the validity and reproducibility of experiments. By maintaining continuity, researchers can test and verify their hypotheses and theories, leading to a better understanding of the natural world.

4. What are some examples of continuity leading to closure in scientific studies?

Examples of continuity leading to closure in scientific studies include maintaining a consistent environment or control group in an experiment, using standardized methods and procedures, and accurately recording and analyzing data throughout a study.

5. How does closure relate to continuity in scientific research?

Closure is the desired endpoint or final state that continuity aims to achieve. Without continuity, achieving closure would be difficult or impossible. Continuity helps to guide the research process and ensure that the desired closure is reached in a logical and consistent manner.

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