Let a b R be a nondegenerate closed bounded interval and let

Let [a, b] R be a non-degenerate closed bounded interval, and let f, g: [a, b] rightarrow R be functions. Suppose that f is zero except at one point Prove that f is integrable and that integral_a^b f(x)dx = 0.

Solution

Let us assume there exists a point c in the interval [a,b], where the value of the function c is non-zero

Let us assume that the value of the function at point c be p, f(c) = p

Case 1: c is not either equal to a nor equal to b

Delta = min(epsilon/2|k|,|c-a|/2,|b-c|/2}, so the partition P will be {a,c-delta,c+delta,b}, where delta is greater than zero

then we can write

U(P,f) - L(P,f) = 2*delta(|p|) <= epsilon

Case 2: when c is equal to a

Then the partition will become P={a,a+delta,b}

In this case we can write

U(P,f) - L(P,f) = delta(|p) < epsilon

Hence for both the cases, i.e. for p>0, L(P,f) is equal to zero and for p<0, U(P,f) is equal to zero

Hence the integral a to b f(x)dx will be equal to zero