Math Help - Continuous functions

1) Consider the formula f(x)=lim(n-->infinity)((x^n)/(1+x^n)).
Let D={x:f(x) is an element of R}. Calculate f(x) for all x elements of D and determine where f: D-->R is continuous.

2) Let f: D-->R and suppose that f(x) greater than equal 0 for all x elements of D. Define sqrt(f)-->R by (sqrt(f))(x) = sqrt(f(x)). If f is continuous at c elements of D, prove that sqrt(f) is continuous at c.

Originally Posted by noles2188

1) Consider the formula f(x)=lim(n-->infinity)((x^n)/(1+x^n)).
Let D={x:f(x) is an element of R}. Calculate f(x) for all x elements of D and determine where f: D-->R is continuous.

2) Let f: D-->R and suppose that f(x) greater than equal 0 for all x elements of D. Define sqrt(f)-->R by (sqrt(f))(x) = sqrt(f(x)). If f is continuous at c elements of D, prove that sqrt(f) is continuous at c.

We have :f(x)= .................................................. .........................................1

AND now we have the following cases:

1) |x|<1 <===> -1<x<1 ,then .

2) x=1 ,then .


3) x>1 ,then .


4) x =-1 ,then oscillates finitely.

5) x<-1 ,then oscillates infinitely.

And thus by using (1)

For -1<x<1 ,f(x) = 0 and

For x=1 , f(x)=1/2.

Thus D= (-1,1] = (-1,1)U{1}

Next we consider continuity over (-1,1].

Let -1<c<1 ,and consider .

HENCE f is continuous over (-1,1).

But if c=1 ,then f(1)=1/2 ,so we see there is a gap from 0 ,which is the value of f(x) for -1<x<1 , to 1/2 =f(1) and hence the function is not continuous.

To prove that:
Let, ε=1/4, then for all δ>0 choose ,x=0 ,then |f(x)-f(1)|=|f(0)-f(1)|=|0-1/2|= .

Thus .

Now to prove : if ,then ,where ,when x belongs to D.

For that you must consider the : .................................................. ..................................2

Now multiply (2) by and we get:


.

But ,thus :


.................................................. .......................................3

And if we want (3) smaller than ε>0 we must have :

which is possible since f is continuous at c.

Hence sqrt(f) is continuous at c

Originally Posted by noles2188

1) Consider the formula f(x)=lim(n-->infinity)((x^n)/(1+x^n)).
Let D={x:f(x) is an element of R}. Calculate f(x) for all x elements of D and determine where f: D-->R is continuous.

The answer above is incomplete.

That means that .
It also means that is continuous on .

The mistake above is: .

Originally Posted by Plato

The mistake above is: .

If |x|>1 ,then x<-1 or x>1 ,but for x=-2 for example the limit of ;

is not equal to 1

Originally Posted by xalk

the limit of is not equal to 1

Yes it is! The reason is that ; and . So as .