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
.
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
.
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) .