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TIFR 2014 Problem 21 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.

The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta

Let be continuous. Suppose for all .

Then

A. no such function exists

B. there are infinitely many such functions

C. there is only one such function

D. there are exactly two such functions

**Discussion:**

Basically, the question is to find out how many such functions can exist.

Let . That is, is the indefinite integral of .

We know from the fundamental theorem of calculus that:

.

So we have for all .

We will manage this equation as we do in case of differential equations. Except that, we have an inequality here.

for all .

Multiplying both sides by the inequality remain unchanged. This is because .

for all .

Now, .

So we have for all .

Therefore, by taking integral from o to y and by using fundamental theorem of calculus:

for all .

i.e, for all .

Also, note that by definition of (F), (F(0)=0). So we have

for all .

But, since for all , we have Â for all .

So far, we have not used the fact that is a non-negative function. Now we use it. Since for all , therefore by monotonicity of the integral, is an increasing function. This means for all .

By the two inequalities obtained above, we get Â for all .

By the fundamental theorem (again!) we get Â for all .

So there is only one such namely the constant function .

**What is this topic:**Real Analysis**What are some of the associated concept:**Fundamental theorem of calculus, Increasing Function**Book Suggestions:**Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert

TIFR 2014 Problem 21 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India's premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.

The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta

Let be continuous. Suppose for all .

Then

A. no such function exists

B. there are infinitely many such functions

C. there is only one such function

D. there are exactly two such functions

**Discussion:**

Basically, the question is to find out how many such functions can exist.

Let . That is, is the indefinite integral of .

We know from the fundamental theorem of calculus that:

.

So we have for all .

We will manage this equation as we do in case of differential equations. Except that, we have an inequality here.

for all .

Multiplying both sides by the inequality remain unchanged. This is because .

for all .

Now, .

So we have for all .

Therefore, by taking integral from o to y and by using fundamental theorem of calculus:

for all .

i.e, for all .

Also, note that by definition of (F), (F(0)=0). So we have

for all .

But, since for all , we have Â for all .

So far, we have not used the fact that is a non-negative function. Now we use it. Since for all , therefore by monotonicity of the integral, is an increasing function. This means for all .

By the two inequalities obtained above, we get Â for all .

By the fundamental theorem (again!) we get Â for all .

So there is only one such namely the constant function .

**What is this topic:**Real Analysis**What are some of the associated concept:**Fundamental theorem of calculus, Increasing Function**Book Suggestions:**Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert

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