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I am studying from Bartle-Shebert and the book says this-

Let $c$ be the point where $f$ and $g$ are not equal.Then, the Riemann sums of $f$ and $g$ are identical with the exception of atmost two points (in the case $c=x_i=x_{i-1}$ is an endpoint). Then, $$|S(f,P)-S(g,P)|=|\sum(f(x_i)-g(x_i))(x_i-x_{i-1})|\le2(|g(c)|+|f(c)|)||P|| $$

Now I can't understand how the last inequality follows. Also I don't see why there are atmost two points at which the Riemann sums are different. If someone can provide a geometrical explanation of this then it would be very helpful.

Note: $P$ is a tagged partition of the interval $[a,b]$ of our consideration and $\{x_0,x_1, ...,x_n\}$ is the partion of $[a,b]$.

PS: I am uploading a picture of the page of the book for reference. enter image description here

Soham
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1 Answers1

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Since $S(f,P) = \sum_{j=1}^n f(\xi_j)(x_j - x_{j-1})$ and $S(g,P) = \sum_{j=1}^n g(\xi_j)(x_j - x_{j-1})$ where $\xi_j \in [x_{j-1},x_j]$, the difference of sums is zero unless $c$ is an intermediate point for both $f$ and $g$ where $f(c) \neq g(c)$.

If $c = \xi_j \in (x_{j-1} , x_j)$ then only one subinterval has a non-vanishing contribution to the difference

$$|S(f,P) - S(g,P)| = |f(c) - g(c)|(x_j - x_{j-1}) \leqslant (|f(c) + |g(c)|) \|P\|$$

The only other cases where the two sums are not identical are where $c = \xi_j = x_j \text{ or } x_{j-1}$ for some $j$ and here is where the factor of $2$ is introduced in the inequality.

For example, if $c = \xi_j = x_j$, then $$|S(f,P) - S(g,P)| = |(f(c) - g(c))(x_j - x_{j-1}) + (f(c) - g(c))(x_{j+1} - x_{j})| \\ \leqslant (|f(c) + |g(c)|)(x_j - x_{j-1}) + (|f(c) + |g(c)|)(x_{j+1} - x_{j}) $$

RRL
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  • If we use the same tagged partitions for both $f$ and $g$ then why do we use $f(\xi)$ and $g(\eta)$ in the Riemann Sums. Shouldn't the tags be same for both? – Soham Mar 18 '20 at 05:12
  • For the second case, is $c$ the tag for two consecutive intervals $[x_j,x_{j-1}]$ and $[x_{j+1},x_{j}]$? – Soham Mar 18 '20 at 05:21
  • I meant them to be the same tagged partition so I can change that notation if you like. – RRL Mar 18 '20 at 05:28
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    Yes, $c$ could possibly be the tag for two consecutive intervals which would allow two sub intervals to have a non-zero contribution to the difference of sums. – RRL Mar 18 '20 at 05:35