Suppose that $f$ is continuous on $[a,b]$. Prove that given $\epsilon>0$, there exist points $x_0=a $E_k=\{y: f(x)=y\ for\ some\ x \in [x_{k-1}, x_k]\}$, then sup$E_k -$inf$E_k<\epsilon$ for $k=1,2,...,n$. Just from looking at the question, I suspect that I need to eventually the Bolzano Weierstrass Theorem and Extreme Value Theorem to complete this proof. However, I am having some difficulty setting up the problem so that I can use theorems as needed. Please help! I really appreciate any guidance that I can get.
Continuity and Subsequences
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analysis
continuity
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0Solving the proof???????????????? How is this possible? – 2012-10-28
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1@LJym89,not everyone speaks English as their first language. – 2012-10-28
1 Answers
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Notice that what the question asks is that continuous functions are Riemann-integrable.
If you have the theorem that states that continuous functions defined on compact sets are uniformly continuous, then it suffices you to choose $$ a = x_0 < x_1 < \dots < x_n = b $$ such that $$ |x_{i+1} - x_i| < \delta, i = 0, 1, \dots, n-1 $$ where $$ \forall \varepsilon > 0,\quad \exists \delta > 0,\quad \forall x,y \in [a,b], \quad |x-y| < \delta \quad \Rightarrow \quad |f(x) - f(y)| < \varepsilon, $$ i.e. the definition of uniform continuity. I believe this is a sufficiently good hint, if you want more details you can ask me again.
Hope that helps,
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0Thank you so much! The hint was actually really helpful and led me in exactly the right direction. I combined the Extreme Value Theorem and the theorem that you stated (which I did know), and I was able to find the proof relatively quickly. Thank you again! – 2012-10-28
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0@Jess : How did you use the EVT? I'm curious because I don't need it. – 2012-10-29
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0I used the EVT to show that because $[x_{k-1},x_{k}]$ is a closed, bounded interval and f is continuous on $[x_{k-1},x_{k}]$, f is bound on $[x_{k-1},x_{k}]$ and $\exists x_{M}, x_{m} \in [x_{k-1},x_{k}] \ni f(x_{M})=M=supE_{k} and f(x_{m})=m=infE_{k}$ Then, I stated that as $[x_{k-1},x_{k}]$ is a closed, bounded interval and f is continuous on $[x_{k-1},x_{k}]$, f is uniformly continuous on that interval. I then used the definition of uniform continuity to show that $|f(x_{M})-f(x_{m})|=|sup(E_{k})-inf(E_{k})|<\epsilon$ – 2012-11-11
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0The problem with your argument is that the last line works if the $\delta_k$ for uniform continuity is such that $|x_M - x_m| < \delta_k$, which nothing ensures you that that will happen. You need to do the exact same reasoning from the start, but begin with the interval $[a,b]$ instead ; if you choose your partition with $x_{k-1} - x_k < \delta$ for all $k$, then uniform continuity over $[a,b]$ will ensure you that $|\mathrm{sup} - \mathrm{inf}| < \varepsilon/(b-a)$, so that when you compute the Riemann sum, it is smaller than $(b-a) \varepsilon/(b-a) = \varepsilon$. – 2012-11-11