More precisely, all well-defined computable functions are continuous. For another exposition of the same (or similar) results, see also Dan Piponi's post from

More precisely, all well-defined computable functions are continuous. For another exposition of the same (or similar) results, see also Dan Piponi's post from 2008 "What does topology have to do with computability?" at http://blog.sigfpe.com/2008/01/what-does-topology-have-to-do-with.html http://blog.sigfpe.com/2008/01/what-does-topology-have-to-do... Also, as a follow-up, you may be interested in Martin Escardo's 2007 post on the same blog: http://math.andrej.com/2007/09/28/seemingly-impossible-functional-programs/ http://math.andrej.com/2007/09/28/seemingly-impossible-funct...