Equivalence of Cauchy integral with Riemann integral

Let $f(x) = x$ on the interval $[0,1]$ and $P = (0,1)$.

Given any refinement $Q = (x_0,x_1, \ldots, x_{n-1},x_n)$ we have, since $f$ is increasing,

$$U(f,P) - C(f,Q) = 1 - \sum_{k=1}^n x_{k-1}(x_k - x_{k-1}) > 1 - \int_0^1x \,dx = 1/2.$$

Take $\epsilon < 1/2$ and we see that the conjectured result cannot be true.

Tags:

Real Analysis

Riemann Integration

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