Distances on Functions

Above we discussed how we may associate a continuous function to a real world phenomenon. Now we explore what we can do with a continuous function or an approximation thereof and the conclusions we can draw from this about the original phenomenon. Let us assume we have two continuous functions \(f \colon X \rightarrow {\mathbb{R}}\) and \(g \colon Y \rightarrow {\mathbb{R}}\) with approximations \(f' \colon X \rightarrow {\mathbb{R}}\) respectively \(g' \colon Y \rightarrow {\mathbb{R}}\).