Real Analysis Answers

Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:

f ={-x², when x ≤ 0.

4-5x , when 0<x≤1

3x-4x², when 1<x≤2

-12x + 2x , when x>2}

Also check whether the function f is derivable at x = 1

Evaluate:

Lim↓ n→∞ [ n/1+n² + n/4+ n² +n/9+ n² +...... +n/2n²]

Show that (1/n²+ n+1)↓n∈N

is a Cauchy sequence.

The product of two divergent sequences is divergent. True or false? Justify.

Give an example of a divergent sequence which has two convergent subsequences.

Justify your claim.

Give an example for each of the following:

A bijection from N↓odd to Z.

Give an example for each of the following: _

A set S with S° = S.

Give an example for each of the following:

A set having no limit point.

Give an example for each of the following:

A set in R whose all points except the one are its limit points.