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Math Help - Continuity

  1. #1
    mms
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    Continuity

    Let f be a continous function in [a,b] and
    <br />
g(x) = \left\{ \begin{gathered}<br />
f(a)\,\,\,x = a \hfill \\<br />
\max \left\{ {f(x):\,x \in \left[ {a,x} \right]} \right\}\,\,\,x \in (a,b] \hfill \\ <br />
\end{gathered} \right.<br />

    prove that g is continous in [a,b]
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  2. #2
    mms
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    hmm well the problem says that g(x)= max f(x) where x is in [a,x] :/
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  3. #3
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by mms View Post
    hmm well the problem says that g(x)= max f(x) where x is in [a,x] :/
    Yes sorry, I misread this part.
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  4. #4
    ynj
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    Quote Originally Posted by mms View Post
    Let f be a continous function in [a,b] and
    <br />
g(x) = \left\{ \begin{gathered}<br />
f(a)\,\,\,x = a \hfill \\<br />
\max \left\{ {f(x):\,x \in \left[ {a,x} \right]} \right\}\,\,\,x \in (a,b] \hfill \\ <br />
\end{gathered} \right.<br />

    prove that g is continous in [a,b]
    \forall\epsilon>0,\exists|\delta|>0,\forall|h|<\de  lta,|f(x+h)-f(x)|<\epsilon.
    since g(x+h)=\max\{g(x-|h|),max\{f(x)|x\in[x-|h|,x+|h|\}\}, so if g(x+h)=f(y) for some y\in[a,x-|h|], then g(x+h)-g(x)=0,if g(x+h)=f(y)for some y\in[x-|h|,x+|h|],then |g(x+h)-g(x)|=|f(y)-g(x)|\leq|f(y)-f(x)|\leq\epsilon.In any case, |g(x+h)-g(x)|\leq\epsilon. So g(x)\in C[a,b].
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