Question

In: Advanced Math

Let φ : R −→ R be a continuous function and X a subset of R...

Let φ : R −→ R be a continuous function and X a subset of R with closure X' such that φ(x) = 1 for any x ∈ X. Prove that φ(x) = 1 for all x ∈ X.'

Expert Solution

Let be a continuous function

and with closure such that

.

Prove that :

Solution :

with closure

Therefore every point of is a point of closure of .

So by definition of point of closure

if , then there exists a sequence of points in such that .

we have to prove that .

So we let be any arbitrary element of ,

Now we will prove that

Because is closure of , therefore there exists a sequence of points in such that .

So and

Also is a continuous function .

So in particular is continuous at .

Therefore by definition of continuity at a point .

if then

Now , therefore ............. ( because )

Hence is a constant sequence and it converges to .

Also .

But a convergent sequence converges to a unique limit .

Therefore .

So we have proved that for any arbitrary .

Therefore

for every .

Therefore we proved

Colby Messinger answered 2 years ago

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