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Thread: Equicontinuity Question

  1. #1
    Member Haven's Avatar
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    Equicontinuity Question

    Let $\displaystyle f$ be a continuous function on $\displaystyle \mathbb{R}$. Let $\displaystyle f_n = f(nt)$ for $\displaystyle n \in \mathbb{N}$ be equicontinuous on [0,1]. I.e., $\displaystyle \forall \epsilon >0, \exists \delta >0 $ such that if $\displaystyle |x-y| < \delta$, then $\displaystyle |f_n(x) - f_n(y)| < \epsilon$ for all n.

    What can we conclude about $\displaystyle f$?

    All I am able to get is that since each $\displaystyle f_n$ is defined on a compact set, then each $\displaystyle f_n$ is pointwise bounded. So, $\displaystyle \{f_n \}$ is uniformly bounded. So, $\displaystyle f$ is uniformly bounded.

    Is there something else that I can conclude?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Haven View Post
    Let $\displaystyle f$ be a continuous function on $\displaystyle \mathbb{R}$. Let $\displaystyle f_n = f(nt)$ for $\displaystyle n \in \mathbb{N}$ be equicontinuous on [0,1]. I.e., $\displaystyle \forall \epsilon >0, \exists \delta >0 $ such that if $\displaystyle |x-y| < \delta$, then $\displaystyle |f_n(x) - f_n(y)| < \epsilon$ for all n.

    What can we conclude about $\displaystyle f$?

    All I am able to get is that since each $\displaystyle f_n$ is defined on a compact set, then each $\displaystyle f_n$ is pointwise bounded. So, $\displaystyle \{f_n \}$ is uniformly bounded. So, $\displaystyle f$ is uniformly bounded.

    Is there something else that I can conclude?
    You can show that $\displaystyle f$ must be constant. Can you prove that?
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  3. #3
    Member Haven's Avatar
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    Quote Originally Posted by Drexel28 View Post
    You can show that $\displaystyle f$ must be constant. Can you prove that?
    Actually, I am having a bit of difficult getting showing it is constant. I consider $\displaystyle x \in \mathbb{R}$ and let $\displaystyle f(x) = c$. Now I suppose $\displaystyle \exists y\in \real$ such that $\displaystyle f(y) \neq c$. Now let $\displaystyle n$ be a natural number larger than y. It follows that $\displaystyle f_n(y/n)= f(y) \neq c$.

    However, I am having troubles with the $\displaystyle \delta$'s in this case. for a given epsilon, Does the intermediate value theorem guarantee that I can find $\displaystyle z$ such that $\displaystyle x<z<y$ and $\displaystyle z-x < \delta$?
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  4. #4
    Super Member girdav's Avatar
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    You can write that $\displaystyle |f(x)-f(y)| =|f_n\left(\dfrac xn\right)-f_n\left(\dfrac yn\right)|$. Take a $\displaystyle \varepsilon >0$. We can find a $\displaystyle n$ such that $\displaystyle \left|\dfrac xn-\dfrac yn\right|\leq \delta$ hence $\displaystyle |f(x)-f(y)|\leq\varepsilon$. Now you can conclude.
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