Continuous functions are integrable proof

How to use

• Visualize the proof idea: for every $\varepsilon>0$ we construct staircase functions $\psi$ and $\varphi$ such that $\psi \le f \le \varphi \quad\text{and}\quad I(\varphi)-I(\psi)=\varepsilon.$
• The applet shows when the key “mesh-size condition” of the proof is satisfied.
• Drag the blue points $a$ and $b$ to choose the interval $[a,b]$.
• Use the slider $\varepsilon$ to choose the tolerance.
• Use the slider $n$ to choose the number of subintervals in the uniform partition $Z$.
• The blue curve is the continuous function $f$.
• The partition $Z$ is uniform with mesh size $\Delta x=\frac{b-a}{n}.$
• The applet computes $\tilde\varepsilon=\frac{\varepsilon}{2(b-a)}.$ The vertical bracket at $x=b$ has height $2\tilde\varepsilon$.
• The orange staircase is $\varphi$ (intended upper bound), defined on each open interval $(x_{i-1},x_i)$ by $\varphi(x)=f(x_i)+\tilde\varepsilon.$
• The purple staircase is $\psi$ (intended lower bound), defined on each open interval $(x_{i-1},x_i)$ by $\psi(x)=f(x_i)-\tilde\varepsilon.$
• The filled band between them is colored green/red depending on whether the proof’s size condition is satisfied.
• The proof requires choosing $n$ so that $\Delta x < \delta$, where $\delta$ comes from uniform continuity (condition (7.2)).
• In the applet, this is checked via a sufficient bound (using an estimate of $\sup|f'|$ for the fixed function $f$). Concretely, it computes a number $\delta$ and checks whether $\frac{b-a}{n} < \delta.$
• If the applet shows the red message “$(b-a)/n \ge \delta$”, then the proof hypothesis is not satisfied.
  • In this red case, $\psi$ and $\varphi$ are still drawn, but they are not guaranteed to satisfy $\psi \le f \le \varphi$ on every subinterval.
  • That is exactly why you sometimes see the purple staircase above the blue curve, or the orange staircase below it: the construction only becomes a true lower/upper bound after the mesh is fine enough.
• To make $\psi\le f\le\varphi$ hold, you need to reach the green state. You can do that by:
  • increasing $n$ (smaller $\Delta x$),
  • shrinking the interval (smaller $b-a$),
  • or increasing $\varepsilon$ (larger $\tilde\varepsilon$, hence an easier condition).
• Once the display turns green, the intended implication of (7.2) is in effect:$|x-x_i|<\delta \Rightarrow |f(x)-f(x_i)|<\tilde\varepsilon,$ which gives $\psi(x)\le f(x)\le\varphi(x)$ on each $(x_{i-1},x_i)$, matching the proof idea.

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Theorem7.5 page 129

This applet is quiet experimental and visualizes the proof that a continuous function $f:[a,b]\to\mathbb{R}$ is Riemann integrable by constructing two staircase functions $\psi\le f\le\varphi$ whose integrals differ by at most $\varepsilon$. The function is fixed in the code as const f = (x: number) => 0.5 * Math.sin(x) * x + 2.5;.

A good way to use the applet is the same order as the proof. First fix an interval $[a,b]$ by dragging the gliders gliderA and gliderB. Then choose an error tolerance $\varepsilon>0$ with epsSlider. After selecting the $\varepsilon>0$, thge student should decrease the interval or add more n-steps nSlider, to get condition 7.2 ($(b-a)/n < \delta$) works and therefore the all construction.

The applet computes $\widetilde{\varepsilon}=\varepsilon/(2(b-a))$ via const eTilde = eps / (2 * range);. The dashed bracket at $x=b$ has vertical length $2 \widetilde{\varepsilon}$ (objects epsBracket and epsBracketLabel), and it is colored green/red together with the construction.

Next choose $n$ (the number of subintervals) with nSlider. The applet creates the uniform partition $Z: a=x_0<\dots<x_n=b$ with $x_k=a+k(b-a)/n$, implemented by uniformPartition(a,b,n), where dx=(b-a)/n is the mesh size, as in the proof.

const uniformPartition = (a: number, b: number, n: number) => {
    const dx = (b - a) / n;
    const cuts = Array.from({ length: n + 1 }, (_, k) => a + k * dx);
    cuts[cuts.length - 1] = b;
    return { cuts, dx };
};

On each interval $(x_{i-1},x_i)$ the staircase functions are defined using the right endpoint $x_i$: upperH[i] = f(x_i) + eTilde; and lowerH[i] = f(x_i) - eTilde; ( I haven’t included codition 7.3: $\psi(x_i) = \varphi(x_i) = f(x_i)$).

These are drawn as step graphs (upperStep for $\varphi$ in orange and lowerStep for $\psi$ in purple). The filled region between them (bandPoly) illustrates the intended inequality $\psi \le f \le \varphi$.

To make the $\delta$-step (7.2) explicit, the code uses a Lipschitz bound derived from $f'(x)$. Since $f(x)=\tfrac12 x\sin x + 2.5$, we have $f'(x)=\tfrac12(\sin x + x\cos x)$ and therefore $|f'(x)|\le \tfrac12(1+|x|)$. Hence on $[a,b]$ we can take supDf = 0.5 * (1 + max(|a|,|b|)) and define delta = eTilde / supDf.

// δ ensuring (7.2)
const maxAbsX = Math.max(Math.abs(a), Math.abs(b));
const supDf = 0.5 * (1 + maxAbsX);
const delta = eTilde / supDf;
// dx < δ
deltaCopndition = dx < delta;

When dx < δ holds, condition (7.2) guarantees that for all $x\in(x_{i-1},x_i)$ we have $|f(x)-f(x_i)|<\widetilde{\varepsilon}$, which implies $\psi(x)\le f(x)\le\varphi(x)$ on each subinterval. The band (and the $\widetilde{\varepsilon}$-bracket) turns green when this condition is satisfied, and red when it is not guaranteed.