Lipschitz Continuity

Accumulation Point:

Let $A \subseteq \mathbb{R}$ be a set and $a \in \mathbb{R}$ . Then the following statements are equivalent: Let $D \subset \mathbb{R}$ and $f : D \to \mathbb{R}$ . The function $f$ is called Lipschitz continuous if there exists an $L > 0$ such that for all $x, y \in D$ it holds that $|f(x) - f(y)| \le L\, |x - y|$ .

How to use the applet

Drag the two orange points on the curve: they represent $(x_1,f(x_1))$ and $(x_2,f(x_2))$ .
Move the slider $L$ to change the allowed Lipschitz constant.
The red dashed line is the secant line through the two chosen points.
The right triangle encodes the differences $\Delta x$ (horizontal) and $\Delta y$ (vertical), so the (absolute) secant slope is

\frac{|\Delta y|}{|\Delta x|}=\frac{|f(x_2)-f(x_1)|}{|x_2-x_1|}.

The arrows labeled $+L$ and $-L$ visualize the slope bounds $\pm L$ .
The shaded “cone” with vertex at the left point shows the region between the two lines of slope $\pm L$ .
Green: the Lipschitz inequality holds for the currently chosen points (the secant slope is $\le L$ in absolute value).
Red: the inequality fails for the chosen points (you would need a larger $L$ ).
For $f(x)=\sin x$ , choosing $L\ge 1$ makes the condition succeed for all pairs of points (consistent with the Mean Value Theorem and $|\cos z|\le 1$ ).

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Hold Shift + scroll to zoom
Hold Shift + drag to move
Points shaped like <> act as sliders

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Definition 4.7 page 65

This applet represents the standard geometric interpretation of Lipschitz continuity.

Given the condition that for an $L > 0$ $|f(x) - f(y)| \le L,|x - y|$

and that the slope of a linear function is “slope = vertical difference divided by horizontal difference,” the slope of the secant connecting the points $f(x)$ and $f(y)$ has slope $\frac{|f(x) - f(y)|}{|x - y|}$ . From the $L$ -condition we retrieve $\frac{|f(x) - f(y)|}{|x - y|} \le L$ . The applets updates the color accordingly:

const isConditionMet = () =>
    Math.abs(p2.Y() - p1.Y()) <=
    lSlider.Value() * Math.abs(p2.X() - p1.X());

const color = () => (isConditionMet() ? COLORS.lightGreen : COLORS.lightRed);

Thus the absolute slope is constrained between $\pm L$ , as shown.

I am aware that this touches on the concept of slope, not yet covered by the student at this point, and that this connection is even better explained using the MVT.

As a simple example I chose $\sin$ , whose derivative is $\cos$ . By the MVT we get that $\frac{f(x) - f(y)}{x - y} = \cos(z)$ for some $z$ between $x$ and $y$ . But $|\cos(z)| \le 1 = L > 0$ .