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Thread: connected set

  1. #1
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    connected set

    let $\displaystyle f(x)$be continuous on $\displaystyle [ 0,\infty )$. Define $\displaystyle L=\{z\in R \ni \exists$ $\displaystyle a$ $\displaystyle sequence$ $\displaystyle x_n \rightarrow \infty$ $\displaystyle and$ $\displaystyle f(x_n) \rightarrow z$ $\displaystyle as$ $\displaystyle n \rightarrow \infty\}$
    Prove a) $\displaystyle L$ is closed b)$\displaystyle L$ is connected c)$\displaystyle L$ is an interval
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  2. #2
    Super Member Rebesques's Avatar
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    If you recall the definitions of limes superior and limes inferior, prove $\displaystyle L=[{\rm liminf}_{\infty}f,{\rm limsup}_{\infty}f]$.
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