Metric spaces, bounded variation

If f is a continuous mapping of a compact metric spaces X into a metric spaces Y, then

• f is uniformally continuous

• f is not continuous

• f has a jump at x=0

• f is a step function

Which of the following(s) is / are true?

• The compact subset of a metric spaces is closed

• The compact subset of a metric space is open

• The compact subset of a metric space is bounded

• The compact subset of a metric space is unbounded

If f and g are two measurable real valued functions defined in same domain, then

• f+g is measurable

• f+g is not measurable

• fg is measurable

• fg is not measurable

If A is an open set and B is a closed set in Rⁿ, then

• B-A is closed set

• B-A is open set

• B-A is null set

• B-A is semi open set

Which one of the following is true?

• The continuous image of a compact set is compact

• The continuous image of a compact set is not compact

• Both a and b

• Neither a nor b

A continuous function

• Is always a function of bounded variation

• Is never a function of bounded variation

• May or may not be a function of bounded variation

• None of the above

Let F be an open covering of A, then

• There exist countable sub collection of F which covers A

• There exist uncountable sub collection of F which covers A

• Both a and b are true

• Both a and b are false

A metric space X is compact

• It is sequentially compact

• It is not sequentially compact

• It doesn't satisfy bolzano weierstrass property

• None of the above

If A is closed subset of complete metric space, then

• A is complete metric space

• A is incomplete metric space

• Undefined

• None of the above

A metric space (X,ρ) is complete, if

• Every sequence in X is convergent

• Every sequence in X is divergent

• Every cauchy sequence in X is convergent

• Every cauchy sequence in X is divergent