Portsmouth, the car, and the intermediate value theorem

I took L back to Portsmouth at the weekend.

At 1000 the odometer read 99951, and at 1800 it read 100310.

Mathematically the odometer is a function of time, and we can refer to time as the ‘interval’ from 1000 to 1800.

The intermediate value theorem says that if f is a continuous function on the interval [a,b], and u is a number between f(a) and f(b), then there is a c in the interval [a,b] such that f(c) = u.

It’s given that:
f(1000) = 99951
f(1800) = 100310

We choose u – which can be any intermediate value between 99951 and 100310 – to be 99999.

The theorem tells us that there was a time c, between 1000 and 1800, such that f(c) = 99999.

There’s nothing too remarkable here, it’s an intuitive result about how continuously-varying quantities behave. Either way I’m pleased to report that the odometer did indeed – and most pleasingly – pass through 99999 on its way to 100000.

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