Mathematics and Computation

CRANK

27 March 2006Andrej Bauer, ,You may have heard at times that there are mathematicians who think that all functions are continuous. One way of explaining this is to show that all computable functions are continuous. The point not appreciated by many (even experts) is that the truth of this claim depends on what programming language we use.Surely they are not all continuous!?You must be thinking to yourself: how can anyone in their right mind claim that all functions are continuous – here’s one that isn’t:$$\mathrm{sgn}(x)=\begin{cases}-1 & x < 0 \0 & x = 0 \1 & x > 0\end{cases}$$At $x=0$ the sign function jumps from $-1$ to $0$ to $1$, which is a nice discontinuity. As crazy as it seems, it makes sense to refuse to admit that $\textrm{sgn}$ is a legitimate function!The official definition nowadays is that a function $f : A \to B$ is the same thing as a functional relation on $A \times B$. Recall that a relation $R$ on $A \times B$ is just a subset of $A \times B$. It is functional w…