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# ISI MStat 2016 (Sample) Problem 2 | Continuous function | PSB This is a beautiful sample problem from ISI MStat 2016 PSB Problem 2. This is based on application of continuity and integration.

## Problem- ISI MStat 2016 Problem 2

Let be a continuous function. Suppose that exists and for all . If and prove that .

## Solution

Given function is continuous and exists .

Hence we can integrate .

Another thing is given that for all ----(1) (say)

Now if we integrate both side of (1) from -1 to y , where then we get , , for all  ,for all [ since given ]---(2)

Again if we integrate both side of (1) from y to 1 ,where then we get, , for all  ,for all [ since given ] ---(3)

Hence from (2) & (3) we get for all and .

Therefore , for all ( proved )

## Challenge Problem be continuous function such that for every x and Then find f(x) .

This is a beautiful sample problem from ISI MStat 2016 PSB Problem 2. This is based on application of continuity and integration.

## Problem- ISI MStat 2016 Problem 2

Let be a continuous function. Suppose that exists and for all . If and prove that .

## Solution

Given function is continuous and exists .

Hence we can integrate .

Another thing is given that for all ----(1) (say)

Now if we integrate both side of (1) from -1 to y , where then we get , , for all  ,for all [ since given ]---(2)

Again if we integrate both side of (1) from y to 1 ,where then we get, , for all  ,for all [ since given ] ---(3)

Hence from (2) & (3) we get for all and .

Therefore , for all ( proved )

## Challenge Problem be continuous function such that for every x and Then find f(x) .

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