Year 12s: what will be the first way you research your future options? >>

Continuous functions and infinimum

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

|x-a|<\delta

.

So,

|3-g(inf(S))|<\epsilon

|x-inf(S)|<\delta

.

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

Thanks.

(edited 12 years ago)

Reply 1

Raiden10

Original post by RamocitoMorales

So if g is continuous, for any

\epsilon >0

, there exists

\delta >0

such that

|g(x)-g(a)|>\epsilon

|x-a|<\delta

.

So,

|3-g(inf(S))|<\epsilon

|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.

Reply 2

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.

Reply 3

RamocitoMorales

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.

(edited 12 years ago)

Spoiler

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?

(edited 12 years ago)

Huh?!?

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.