**(From I.S.I. M.Math Subjective Sample Paper 2013)**

Let N>0 and let be denoted by f(x) = 1 if x=1/i for some integer and f(x) = 0 for all other values of x. Show that f is Riemann Integrable.

**Discussion**

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 .

S(f,P) be the Riemann Sum of function f given this tagged partition; that is

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 for any (that is we will be able to find a which is the norm of a partition concerned)

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

**Proved**

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