How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
Learn More

Riemann Integrable function

This is a problem from I.S.I. M.Math Subjective Sample Paper 2013 based on Riemann Integrable Function. Try out this problem.

Problem: Riemann Integrable function

Let N>0 and let \mathbf{ f:[0,1] to [0,1] } be denoted by f(x) = 1 if x=1/i for some integer \mathbf{i\le N} and f(x) = 0 for all other values of x. Show that f is Riemann Integrable.


First let's get the notations in place (Riemann integral has several notations in different books).
Let P be a tagged partition of [0,1] that is \mathbf{wp = {([x_{i-1} , x_i ], t_i)}_{i=1}^n }.

S(f,P) be the Riemann Sum of function f given this tagged partition; that is  \mathbf{ S(f, wp) = \sum_{i=1}^n f(t_i)(x_i -x_{i-1}) }

We conjecture that the Riemann Integral of the given function is 0 (how do we know it? A guess. If we wish to eliminate this guessing step, then we have to use Cauchy criterion for the proof).

We show that \mathbf{ S(f, wp) < \epsilon} for any \mathbf{ \epsilon > 0 } (that is we will be able to find a \mathbf{\delta_{\epsilon}} which is the norm of a partition concerned)

Let us take \mathbf{\delta_{\epsilon} = \frac{\epsilon}{2N} } that is we divide [0,1] into \mathbf{\lfloor \frac{2N}{\epsilon} \rfloor } parts of equal length. The Riemann sum of the given function over this partition is at most \mathbf{\frac{\epsilon}{2} } which is smaller than $latex \mathbf{\epsilon}$


Some Useful Links:

Our ISI CMI Entrance Program

Sequence Problem | ISI Entrance B.Math 2008 Obj 1 - Video

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.