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# Telescopic Continuity | ISI MStat 2015 PSB Problem 1 This problem is a simple application of the sequential definition of continuity from ISI MStat 2015 PSB Problem 1 based on Telescopic Continuity.

## Problem- Telescopic Continuity

Let be a function which is continuous at 0 and Also assume that satisfies the following relation for all : Find .

## Solution     Add them all up. That's the telescopic elegance. Observe that since, is continuous at .

Hence take limit on , and we get .

## Food for Thought

• Find the general function from the given condition.
• and g(x) is continuous, then prove that .
• What if is not continuous?

This problem is a simple application of the sequential definition of continuity from ISI MStat 2015 PSB Problem 1 based on Telescopic Continuity.

## Problem- Telescopic Continuity

Let be a function which is continuous at 0 and Also assume that satisfies the following relation for all : Find .

## Solution     Add them all up. That's the telescopic elegance. Observe that since, is continuous at .

Hence take limit on , and we get .

## Food for Thought

• Find the general function from the given condition.
• and g(x) is continuous, then prove that .
• What if is not continuous?

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