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Math Help - Left continuity

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

    If 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:

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

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

    Let r_n=f(a+\epsilon_n).

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

    therefore \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 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:

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

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

    Let r_n=f(a+\epsilon_n).

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

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

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

    Let r_n=f(a+\epsilon_n).

    Since f is an increasing sequence r_1>r_2>r_3.......

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

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

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

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

    Let r_n=f(a+\epsilon_n).

    Since f is an increasing sequence r_1>r_2>r_3.......

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

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

    therefore \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  r_n = f(a-\epsilon_n) . Then  r_n is an increasing sequence. And so  r_{\infty} = f(a) . Now taking  r = a, \ c = c_1 in the Intermediate Value Theorem, we get  f(a) = c_1 . So  \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 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:

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

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

    Let r_n=f(a+\epsilon_n).

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

    therefore \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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