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# Continuous functions and infinimum watch

1. Let be a continuous function.
Let S be a non-empty subset of [0,1].
Suppose for all . Let .
Show that .
So if g is continuous, for any , there exists such that if .

So, if .

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

Thanks.
2. (Original post by RamocitoMorales)
So if g is continuous, for any , there exists such that of .

So, if .

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

Thanks.
Perhaps it's a wrong question.
3. 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.
4. (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.
5. Spoiler:
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Spoiler:
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Spoiler:
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Bump.
6. (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 for all , then . How can I use sequential continuity to show that though?
7. Huh?!?
8. (Original post by RamocitoMorales)
So applying that logic to the question would mean that if for all , then . 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.
9. 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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