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    Let g:[0,1]\to\mathbb{R} be a continuous function.
    Let S be a non-empty subset of [0,1].
    Suppose g(x)=3 for all x\in S. Let a=inf(S).
    Show that g(a)=x.
    So if g is continuous, for any \epsilon >0, there exists \delta >0 such that |g(x)-g(a)|<\epsilon if |x-a|<\delta.

    So, |3-g(inf(S))|<\epsilon if |x-inf(S)|<\delta.

    I'm stuck as to what to do now? Any help would be appreciated.

    Thanks.
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    (Original post by RamocitoMorales)
    So if g is continuous, for any \epsilon >0, there exists \delta >0 such that |g(x)-g(a)|>\epsilon of |x-a|<\delta.

    So, |3-g(inf(S))|<\epsilon if |x-inf(S)|<\delta.

    I'm stuck as to what to do now? Any help would be appreciated.

    Thanks.
    Perhaps it's a wrong question.
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    Using sequential continuity you can show that if
    g(y)=a for all y in S then g(inf(S))=a
    I think thats what the questions asking. your definition of continuity has a typo.
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    (Original post by jj193)
    Using sequential continuity you can show that if
    g(y)=a for all y in S then g(inf(S))=a
    I think thats what the questions asking. your definition of continuity has a typo.
    Okay, I've fixed the typing errors.
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    (Original post by jj193)
    Using sequential continuity you can show that if
    g(y)=a for all y in S then g(inf(S))=a
    I think thats what the questions asking. your definition of continuity has a typo.
    So applying that logic to the question would mean that if g(x)=3 for all x\in S, then g(inf(S))=3. How can I use sequential continuity to show that though?
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    Huh?!?
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    (Original post by RamocitoMorales)
    So applying that logic to the question would mean that if g(x)=3 for all x\in S, then g(inf(S))=3. How can I use sequential continuity to show that though?
    what if inf(S) is in S?
    what if inf(S) is not in S - but in some sense of distance inf(S) is arbitrarily close to some element(s) of S now use continuity. equivalently inf(S) is a limit point of S, use sequential continuity.

    there are a few equivalent formulations of inf(S), having a grasp of all of them would help. Some are easier to work with then others. In this case knowledge of a sequential formulation is the (and often is) least hassle.
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    just write out the definition for a = lim inf S (for all d>0 there is x in S s.t. a<= x<=a+d) and for continuity of g and stare at it for a bit,
 
 
 

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