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

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

- continuity
- Integration

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 )

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.

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

- continuity
- Integration

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 )

be continuous function such that for every x and

Then find f(x) .

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