*In the criminal justice system, velocity-based offenses are considered especially unimportant. In New York, the dedicated detectives who investigate these minor misdemeanors are members of an elite squad known as the Moving Violation Team. These are their stories.*

[Open with aerial shot of the New York State Thruway. It is a beautiful fall day. Traffic is on the heavy side but moving freely. Zero in on a car passing below a set of cameras on the road.]

*DUN DUN*

[One week later, in Syracuse, NY. MICHELLE ROLLINS is bringing the mail inside. Their wife CARLA GOFF is sitting on the couch.]

MICHELLE ROLLINS: What’s this? [opens envelope] Really? A ticket? But I didn’t see any cops when I was driving last week.

CARLA GOFF: They have those cameras mounted above the roads now.

MICHELLE ROLLINS: I saw those. They were right near the toll plazas. I was never speeding when I was near one of the cameras. This is garbage! They can’t prove I was speeding.

[A few days later, in the Moving Violation Team office. Detective DOROTHY BERNSTEIN is going through papers, filing some and tossing others. Her colleague EDDIE WILLIAMS looks on.]

DOROTHY BERNSTEIN: We’ve got another driver contesting the ticket.

EDDIE WILLIAMS: They just don’t stop, do they? They have no idea what they’re in for.

*DUN DUN*

[Inside the courtroom. Judge CHARLOTTE SCOTT presiding. Another *DUN DUN* for good measure.]

BAILIFF: Please rise.

CHARLOTTE SCOTT: You may be seated. What do we have today? Ah, a contested speeding ticket. Plaintiff, opening statement, please.

MICHELLE ROLLINS: Your honor, I received a speeding ticket, but I was never pulled over.

DOROTHY BERNSTEIN: Are you familiar with the cameras we have to record license plates for tolls?

MICHELLE ROLLINS: Sure.

DOROTHY BERNSTEIN: They also record your location and time.

MICHELLE ROLLINS: Of course, but I don’t see how that’s relevant.

DOROTHY BERNSTEIN: There are multiple cameras. We recorded you driving here, at mile marker 192, [holds up blurry photo of a car passing under a camera on the road] at 12:47 pm on October 12. Then we took this photograph of you at mile marker 148. Can you read the timestamp on that photograph for me?

MICHELLE ROLLINS: Are these theatrics really necessary?

CHARLOTTE SCOTT: Just answer the question.

MICHELLE ROLLINS: It says [squints] 1:21 pm.

DOROTHY BERNSTEIN: In 34 minutes, you traveled 44 miles. Is that correct?

MICHELLE ROLLINS: Yes.

DOROTHY BERNSTEIN: The speed limit for this entire portion of the highway is 65 miles per hour. Would you agree that your average speed was above 65 miles per hour?

MICHELLE ROLLINS: [muttering] 68 minutes, 88 miles, 60 minutes, 65 miles, plus 8 is 73, the extra is less than a mile…[regular voice] Yes, it was.

DOROTHY BERNSTEIN: As a matter of fact, it was 77.65 miles per hour.

MICHELLE ROLLINS: But that doesn’t prove anything. The speed limit is not an average speed limit. You have to show I was traveling above 65 miles per hour at some point.

DOROTHY BERNSTEIN: Mx. Rollins, are you familiar with the Mean Value Theorem?

*DUN DUN*

[But no scene change]

MICHELLE ROLLINS: Yeah, I took calculus. That’s the theorem that says that if your average rate of change between two endpoints is M, then your instantaneous rate of change at some point between two endpoints must have been M, if—

CHARLOTTE SCOTT: [bangs gavel] Case closed!

MICHELLE ROLLINS: Wait a minute, I didn’t finish! That’s *if* the function is a continuous function on the whole closed interval and differentiable on the open interval!

CHARLOTTE SCOTT: Are you saying the function describing your position was somewhere discontinuous or non-differentiable?

MICHELLE ROLLINS: I didn’t say that, but, with all due respect, it’s not my responsibility to prove they weren’t but Detective Bernstein’s to prove they were. Detective Bernstein, can you show that time and position are continuous, rather than discrete, quantities?

DOROTHY BERNSTEIN: Oh, please! Your honor, all widely-used modern and classical physical theories that are used to make predictions about real-world behavior use the assumption of continuous time. If time is not continuous, it is close enough on a practical level to assume such.

MICHELLE ROLLINS: By the same token, though, all numbers can be practically represented — to any degree of accuracy we desire — by rational numbers, can they not?

DOROTHY BERNSTEIN: Objection, your honor, irrelevant.

CHARLOTTE SCOTT: Mx. Rollins, where are you going with this?

MICHELLE ROLLINS: I promise it is highly relevant. At all points of my journey, we can assume the time and my position were rational numbers, using Detective Bernstein’s “close enough on a practical level” criterion. Therefore my position was a function of time defined on the rational numbers. The mean value theorem does not hold for functions defined over the rationals! Take, for example, the function that is 0 for all rational numbers *q* such that *q** ^{2}* is less than 2 and 1 for all rational numbers whose squares are larger than 2. The average value of this continuous function on the [0,2] interval is strictly between 0 and 1, but the function only takes the values 0 and 1.

[A gasp ripples through the courtroom, which somehow is full of an audience of people despite the fact that this is a very boring traffic case.]

CHARLOTTE SCOTT: [bangs gavel] Order! Order! Detective Bernstein?

DOROTHY BERNSTEIN: [Stammering] Wait- I- what- You can’t be serious!

CHARLOTTE SCOTT: If the detective cannot counter Mx. Rollins’ argument, I have no choice but to dismiss the ticket.

[DOROTHY BERNSTEIN sinks into her chair. EDDIE WILLIAMS brings her a cup of coffee. MICHELLE ROLLINS leaves the courtroom to a flock of reporters outside.]

DOROTHY BERNSTEIN: Thanks, Eddie. I can’t believe they’re getting away with it.

EDDIE WILLIAMS: We can only hope the next one hasn’t thought so deeply about the mean value property.

*DUN DUN*

*Inspired by conversations on Twitter, **Patrick Honner’s traffic ticket**, **James Propp’s episode of My Favorite Theorem** and his article “**Real Analysis in Reverse,**” the one (1) episode of Law & Order: SVU I watched on a recent flight, and **this 1966 educational video about the Mean Value Theorem**.*