Distance Between Functions

Real Analysis · Axiom Academy

The Distance Between Two Functions Two curves can be "close." But how close, exactly? Measure the distance between functions, and a whole space of functions opens up — each one a single point. How far apart are two functions? You know how to measure the distance between two numbers, or two points on a plane. But two whole functions ? They live on every input at once. The trick that unlocks real analysis is to find a single number that says how far apart they are — and once you can, every function becomes a single point , and the curves you graph turn into a space you can measure inside of. Here are two functions on the interval [0, 1] : in blue and in red. Watch the caliper sweep across, measuring the vertical gap |f(x) - g(x)| at every x . It remembers the biggest gap it ever sees — that single worst-case gap is the distance . The distance isn't an average or a total — it's the single largest vertical gap anywhere on the interval. That's the uniform , or supremum , metric. Move one function and watch the distance change Keep fixed, and reshape by dragging its amplitude a . The caliper always finds the largest vertical gap, and the distance updates with it. Slide a toward 0 and the two functions — two points in the space — drift closer together. A function is now a single point; is the ruler between points. Push and the distance bottoms out at 1 — even a flat g can't escape f 's peak. The same pair, measured two ways

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