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Thread: Left continuity

  1. #1
    Super Member Showcase_22's Avatar
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    Left continuity

    If $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and satisfies the conclusion of the intermediate value theorem, then prove f is left continuous.

    My attempt:

    $\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

    Hence pick $\displaystyle b=a+\epsilon$ and let $\displaystyle \epsilon=\epsilon_n$ s.t $\displaystyle \epsilon_n=1/n$.

    Let $\displaystyle r_n=f(a+\epsilon_n)$.

    Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

    therefore $\displaystyle \lim_{n \rightarrow \infty} f(a+\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

    Hence it is left continuous.

    Is this right? I have a feeling that i've assumed it to prove it
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  2. #2
    Senior Member Sampras's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    If $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and satisfies the conclusion of the intermediate value theorem, then prove f is left continuous.

    My attempt:

    $\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

    Hence pick $\displaystyle b=a+\epsilon$ and let $\displaystyle \epsilon=\epsilon_n$ s.t $\displaystyle \epsilon_n=1/n$.

    Let $\displaystyle r_n=f(a+\epsilon_n)$.

    Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

    therefore $\displaystyle \lim_{n \rightarrow \infty} f(a+\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

    Hence it is left continuous.

    Is this right? I have a feeling that i've assumed it to prove it
    It seems that you should consider $\displaystyle r_n = f(a-\epsilon_n) $ since you want to show left continuity.
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  3. #3
    Super Member Showcase_22's Avatar
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    $\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

    Hence pick $\displaystyle b=a-\epsilon$ and let $\displaystyle \epsilon=\epsilon_n=1/n$.

    Let $\displaystyle r_n=f(a+\epsilon_n)$.

    Since f is an increasing sequence $\displaystyle r_1>r_2>r_3......$.

    f is bounded below by $\displaystyle f(a)$. Therefore $\displaystyle r_{\infty}=f(a)$.

    Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

    therefore $\displaystyle \lim_{n \rightarrow \infty} f(a-\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

    Hence it is left continuous.

    Is that better?
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  4. #4
    Senior Member Sampras's Avatar
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    Quote Originally Posted by Showcase_22 View Post
    $\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

    Hence pick $\displaystyle b=a-\epsilon$ and let $\displaystyle \epsilon=\epsilon_n=1/n$.

    Let $\displaystyle r_n=f(a+\epsilon_n)$.

    Since f is an increasing sequence $\displaystyle r_1>r_2>r_3......$.

    f is bounded below by $\displaystyle f(a)$. Therefore $\displaystyle r_{\infty}=f(a)$.

    Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

    therefore $\displaystyle \lim_{n \rightarrow \infty} f(a-\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

    Hence it is left continuous.

    Is that better?
    I would let $\displaystyle r_n = f(a-\epsilon_n) $. Then $\displaystyle r_n $ is an increasing sequence. And so $\displaystyle r_{\infty} = f(a) $. Now taking $\displaystyle r = a, \ c = c_1 $ in the Intermediate Value Theorem, we get $\displaystyle f(a) = c_1 $. So $\displaystyle \lim_{n \to \infty} f(a-\epsilon_n) = r_{\infty} = f(a) = c_1 $.
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  5. #5
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    Quote Originally Posted by Showcase_22 View Post
    If $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and satisfies the conclusion of the intermediate value theorem, then prove f is left continuous.

    My attempt:

    $\displaystyle \forall r \in [f(a),f(b)] \exists c \in [a,b] \ s.t \ f(c)=r.$

    Hence pick $\displaystyle b=a+\epsilon$ and let $\displaystyle \epsilon=\epsilon_n$ s.t $\displaystyle \epsilon_n=1/n$.

    Let $\displaystyle r_n=f(a+\epsilon_n)$.

    Also let $\displaystyle f(a)=c_1$ (just to make it different from the definition).

    therefore $\displaystyle \lim_{n \rightarrow \infty} f(a+\epsilon_n)\rightarrow f(r_{\infty})=f(a)=c_1$

    Hence it is left continuous.

    Is this right? I have a feeling that i've assumed it to prove it

    Showcase, the intermediate value theorem says:

    If f(x) is continuous in [a,b] and if f(a) =A and f(b)=B,then corresponding to any number C between A and B there exists at least one number c in [a,b] such that f(c) =C.

    So it seems to me you are trying to prove the basic assumption of the intermediate value theorem,since if a function is continuous then is left,right continuous
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