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This is a problem from ISI MStat 2018 PSA Problem 7 based on Continuity.

Let be a function defined from to such that

and for all x

Then is

- (A) continuous and bounded.
- (B) continuous but not necessarily bounded.
- (C) bounded but not necessarily continuous.
- (D) neither necessarily continuous nor necessarily bounded.

Epsilon-Delta definition of limit

Continuity

Bounded function

Answer: is (A)

ISI MStat 2018 PSA Problem 7

Introduction to Real Analysis by Bertle Sherbert

Try to use the epsilon-delta definition of limit and the property that for all x.

such that . Now for all for every .

Let's try to use this .

Given any we can select a suitable such that . Then . But . Hence , . Hence , for all we have . Since , is arbitrary , we must have for all

So is continuous and bounded

This is a problem from ISI MStat 2018 PSA Problem 7 based on Continuity.

Let be a function defined from to such that

and for all x

Then is

- (A) continuous and bounded.
- (B) continuous but not necessarily bounded.
- (C) bounded but not necessarily continuous.
- (D) neither necessarily continuous nor necessarily bounded.

Epsilon-Delta definition of limit

Continuity

Bounded function

Answer: is (A)

ISI MStat 2018 PSA Problem 7

Introduction to Real Analysis by Bertle Sherbert

Try to use the epsilon-delta definition of limit and the property that for all x.

such that . Now for all for every .

Let's try to use this .

Given any we can select a suitable such that . Then . But . Hence , . Hence , for all we have . Since , is arbitrary , we must have for all

So is continuous and bounded

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