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Suppose that $f : (X, d) \to (Y, e)$ is a map between metric spaces, when trying to prove continuity of $f$ at a point $a \in X$ we let $\epsilon > 0$ and we attempt to find a $\delta_{\epsilon}>0$ such that $d_X(x, y) < \delta_{\epsilon} \implies d_Y(f(x), f(y)) < \epsilon$.

The only real hard part to this proof is finding such a $\delta_{\epsilon}$ for a given $\epsilon$. Usually we do this in practice by working backwards from what we want to get, and then omit our rough calculations from the proof and then prove forwards using our $\delta_{\epsilon}$ as is we pulled it out of thin air.

Similarly when we have a function $f : [a, b] \to \mathbb{R}$ and we want to show that $f$ is Riemann integrable (in the way I learnt it through Darboux sums) we choose a "nice collection" (usually countable and indexed by $n$ at least in the examples I've seen so far) of partitions $\{P_n\}_{n \in \mathbb{N}}$ such that each $P_n$ is the image of some discrete function $g : \{1, ..., n\} \to [a, b]$ such that $g(1) = a$ and $g(n) = b$ and $g(n_1) < g(n_2)$ for $n_1 < n_2 \in \{1, ..., n\}$ so $P_n$ is defined as $P_n = \{g(1), ..., g(n)\}$.

Then if we show that $\inf\{U(f, P_n) \ | P_n \in \{P_n\}_{n \in \mathbb{N}}\} = \sup\{L(f, P_n) \ | P_n \in \{P_n\}_{n \in \mathbb{N}}\}$ we've proven that $f$ is Riemann integrable on $[a, b]$ and that $$\int_{a}^b f(x)dx = \inf\{U(f, P_n) \ | P_n \in \{P_n\}_{n \in \mathbb{N}}\} = \sup\{L(f, P_n) \ | P_n \in \{P_n\}_{n \in \mathbb{N}}\}$$

And also actually computing the $\inf$'s and $\sup$'s become easy because $$\inf\{U(f, P_n) \ | P_n \in \{P_n\}_{n \in \mathbb{N}}\} = \inf_{n \in \mathbb{N}} \{U(f, P_n)\}$$ and $$\sup\{L(f, P_n) \ | P_n \in \{P_n\}_{n \in \mathbb{N}}\} = \sup_{n \in \mathbb{N}} \{L(f, P_n)\}$$

and each upper and lower Darboux sum $U(f, P_n)$ and $L(f, P_n)$ will turn out to be functions of $n$ so taking their $\sup$ and $\inf$'s are relatively straightforward.

Now onto my question. Similarly to the example of continuity of functions between metric spaces, where most of the work goes into finding appropriate $\delta_{\epsilon}$'s, most of the work in proving that a function is Riemann integrable goes into finding a nice collection of partitions.

My question thus is, how does one go about choosing this nice collection of partitions? For example we did a proof in my analysis class to show that the function $f : [0, 1] \to \mathbb{R}$ defined by $f(x) = cx$ for $x \in [0, 1]$ was Riemann integrable. This proof basically relied solely on choosing the collection of partitions $$\{P_n = \{x_i\}_{i=0}^n\}_{n \in \mathbb{N}}$$ where $$x_i = \frac{i}{n}$$ for $i \in \{0, ..., n\}$. After realizing this it's pretty simple to prove that $f$ is Riemann integrable and compute the integral.

Now my question is how to do we know how to choose such nice collections of partitions given that we know $f$? Do people just do this by trial and error (I'm pretty sure they don't). What's the intuition behind choosing such partitions?

Also are there examples of Riemann integrable function for which the "nice collection" of Partitions that we need to prove integrability are uncountable?

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  • $\begingroup$ It's easy enough to use a uniform partition for any increasing (or decreasing) function. But try the infamous function $f(x) = \begin{cases} 1/q, & x=p/q \text{ in lowest terms} \\ 0, & \text{otherwise}\end{cases}$. What partition $P$ gets $U(f,P)<\epsilon$? $\endgroup$ – Ted Shifrin May 10 '18 at 20:28

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