I am studying, "Differential Equations with Applications and Historical Notes, 3rd Edition", by Simmons and Finlay, and came across the following statement:
The value of an integrable function can be redefined at any finite number of points without changing the value of its integral.
I don't understand how this statement could be true.
I would greatly appreciate it if people could please take the time to elaborate on this and convince me that it is true.
P.S. I have an elementary understanding of Riemann integration.
Consider an integral $\int_a^b f(x)\; dx$, where $f$ is Riemann integrable. Let $g(x)$ be the same as $f(x)$ except at finitely many points $p_1, \ldots, p_m$, and $h(x) = g(x) - f(x)$. Thus $$h(x) = \cases{0 & if $x$ is not one of $p_1, \ldots, p_n$\cr \text{something else} & if it is}$$ Let $B = \max_k |h(p_k)|$. So $$ \int_a^b g(x) \; dx = \int_a^b (f(x) + h(x))\; dx = \int_a^b f(x)\; dx + \int_a^b h(x)\; dx$$ But I claim $\int_a^b h(x) \; dx = 0$: if you take a Riemann sum $S = \sum_{j=1}^n h(\xi_j) \Delta x$, at most $m$ terms will be nonzero, and so $ |S| \le m B \Delta x$. As $n \to \infty$, $\Delta x \to 0$ and so $S \to 0$.