If you can prove this, you are a genius.

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In summary, the function f is continuous at a point a in X provided that for each positive number ε there is a positive number δ such that if x is in X and d(a,x)<δ, then p(f(a),f(x))<ε.
  • #1
Jamin2112
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Homework Statement



Are you up to the challenge?

Let (X,d) be a metric space and let A be a nonempty subset of X. Define f:X→ℝ by f(x)=inf{d(x,a): a in A}. Prove that f is continuous.

Homework Equations



Definition. Let (X,d) and (Y,p) be metric spaces. A function f:X→Y is continuous at a point a in X provided that for each positive number ε there is a positive number δ such that if x is in X and d(a,x)<δ, then p(f(a),f(x))<ε. A function f:X→Y is continuous provided it is continuous at each point of X.

The Attempt at a Solution



All I've been able to do so far is understand the function f given in the problem. For example, we could have X = {1, 2, 3}, A = {1}, and d(x,a) = |x-a|; then f(1) = 0, f(2) = 1, and f(3) = 2. BUT THIS STEP FUNCTION ISN'T CONTINUOUS! So do I have to assume f is continuous?
 
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  • #2
Your error is that you have completely ignored the topology, or metric, on X. Since X has only three members, any appropriate metric you assign will give a "discrete topology"- every subset is open. In this particular case, for any [itex]0< \delta< 1[/itex], [itex]d(p,q)<\delta[/itex] implies p= q. In particular, for any [itex]0< \delta< 1[/itex], [itex]d(x, 1)< \delta[/itex] implies x= 1 so that [itex]f(x)= f(1)= 1[/itex] and then [itex]d(f(x), 1)= d(1,1)= 0< \epsilon[/itex].

Yes, this function is continuous. That doesn't require a genius, it only requires knowing the definitions.
 
  • #3
Start by showing the inequality

[tex]\inf_{a\in A}{d(x,a)}\leq d(x,y)+\inf_{a\in A}{d(y,a)}[/tex]
 
  • #4
I hope I'm not wrong but:

the definition does not say that f(x)=d(x,a), it says that [itex]f(x)=inf_{a\in A}d(x,a)[/itex].

So, in your example, f(1)=0 (since 1-1=0) f(2)=0 (since d(x,a) can be either 0 or 1, and the lowest is 1), and f(3)=0 (same reason)...

since [itex]x\in X[/itex] and [itex]a\in A\subset X\Rightarrow a\in X, \forall x,\,\,inf_{a\in A}d(x,a)=0[/itex]

Is that correct?
If so, the proof is straightforward :D
 
  • #5
Thanks for the responses so far! I'll chew on 'em and then come back when I have another question!
 
  • #6
micromass said:
Start by showing the inequality

[tex]\inf_{a\in A}{d(x,a)}\leq d(x,y)+\inf_{a\in A}{d(y,a)}[/tex]

I see ...

So choose x in X, y in A in X, and ε > 0 such that ε > d(x,y).

Since d(a,x) ≤ d(a,y) + d(x,y), we know infa in Ad(a,x) ≤ infa in Ad(a,y) + d(x,y), meaning f(y) ≤ f(x) + d(x,y), or equivalently, f(y) - f(x) ≤ d(x,y).

Thus ε > d(x,y) forces f(y) - f(x) < ε. (This seems wrong.)
 

1. How do you define "genius" in this context?

In this context, "genius" refers to the ability to provide evidence or proof for a scientific theory or concept that has not yet been proven by other scientists.

2. Can anyone become a genius by proving a theory?

While anyone has the potential to make groundbreaking discoveries in the scientific field, becoming a "genius" is not solely determined by proving a theory. It also depends on the significance and impact of the theory on the scientific community.

3. Is there a specific process or method for proving a theory?

There is no one set process or method for proving a theory. It often involves conducting experiments, collecting data, and analyzing results to support the theory. Each scientific field may have its own specific methods for proving theories.

4. Can a theory ever be proven beyond a doubt?

In science, theories are constantly being tested and refined, so there is always a possibility for new evidence to emerge that could challenge or disprove a theory. However, if a theory has been extensively tested and supported by evidence, it can be considered a strong and reliable explanation for a phenomenon.

5. What are some examples of theories that have been proven by geniuses?

Some examples of theories that have been proven by scientists considered geniuses include Albert Einstein's theory of relativity, Charles Darwin's theory of evolution, and Isaac Newton's laws of motion. However, it should be noted that these theories were also supported by the work of other scientists and were not solely the work of one individual.

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