Inequality is exactly the one Rudin uses

  • Thread starter mynameisfunk
  • Start date
  • Tags
    Inequality
In summary, the conversation discusses a function g that is continuous on the interval [0,1], with g(0)=g(1)=0 and satisfies a specific inequality. The goal is to prove that g(x)=0 for all x in [0,1], using a hint involving the maximum of g on the interval. There may be some confusion about the hint and whether it is applicable, as well as some difficulty in proving the statement.
  • #1
mynameisfunk
125
0
suppose that [itex]g:[0,1] \rightarrow \re[/itex] is continuous, [itex]g(0)=g(1)=0[/itex] and for every [itex]c \in (0,1)[/itex], there is a [itex]k > 0[/itex] such that [tex]0 < c-k < c < c+k < 1[/tex] and [tex]g(c)=\frac(1}{2}[/tex][tex] (g(c+k)+g(c-k))[/tex].
Prove that [itex]g(x) = 0[/itex] for all [itex]x \in [0,1][/itex] Hint: Consider sup{[itex]x \in [0,1] | f(x)=M [/itex]} where M is maximum of [itex]f[/itex] on [0,1].




I see that [itex]c=\frac{1}{2}((c+k)+(c-k))[/itex]. I also see that the inequality is exactly the one Rudin uses to prove that the derivative of a local maximum is 0. I don't really understand what the hint is. There is no supremum, right?
What I tried to do and decided I couldn't make it work was to take [itex]\delta > 0[/itex] and take [itex]x_0, x_1[/itex] such that [itex]d(x_1,1)=d(x_0,0)<\delta[/itex] and let [itex]k=d(x_1,1)=(x_0,0)[/itex] so that now [itex]g(c)=0[/itex] when [itex]c=\frac{1}{2}[/itex] and I was going to show that [itex]g(c+k),g(c-k)[/itex] would always have to equal 0, but then I was thinking that what if the function oscillated and intersected the x-axis at 0,1/2, and 1 so that [itex]g(c-k)=-g(c+k)[/itex]. Seems like it would hold for my proof, also I didnt use the hint. HELP! This problem seems easy but I can't seem to wrap my head around it.
 
Last edited:
Physics news on Phys.org
  • #2


Your laytex is sort of screwed up. Take a look. It's not clear what your asking.
 

1. What is the concept of "inequality" in Rudin's work?

"Inequality" in Rudin's work refers to the unequal distribution of resources, opportunities, and power among individuals and groups. It is a social and economic issue that results in disparities and disadvantages for marginalized communities.

2. How does Rudin address inequality in his work?

Rudin addresses inequality by examining its root causes, such as systemic discrimination and unequal access to education and employment. He also advocates for policies and interventions that aim to reduce inequality and promote social justice.

3. What evidence does Rudin use to support his arguments about inequality?

Rudin uses a combination of data, statistics, and case studies to support his arguments about inequality. He also draws on historical and sociological research to demonstrate the long-term effects of inequality on individuals and societies.

4. What impact does inequality have on society?

Inequality has a significant impact on society, leading to social and economic divisions, reduced social mobility, and perpetuation of poverty. It can also contribute to social and political unrest, as well as health and educational disparities.

5. How can we work towards reducing inequality?

Reducing inequality requires a multifaceted approach that addresses both the structural and individual factors contributing to it. This can include implementing policies that promote equal opportunities and access to resources, addressing systemic discrimination, and promoting social awareness and advocacy for marginalized communities.

Similar threads

  • Calculus
Replies
9
Views
2K
Replies
1
Views
793
Replies
1
Views
1K
Replies
5
Views
221
Replies
3
Views
1K
Replies
1
Views
1K
Replies
2
Views
795
  • Calculus
Replies
3
Views
695
Replies
1
Views
952
Back
Top