MAT237Y1 Study Guide - Final Guide: Connected Space, Intermediate Value Theorem, Folland Aircraft
Document Summary
The familiar intermediate value theorem (abbrevi- ated ivt) in 1d applies to a continuous function f whose domain is an interval. To state an analogue of the ivt in higher dimensions, we need to figure out a suitable geometric condition on the domain of a continuous function. A condition that is easy to state, intuitively reasonable, and good enough for all our purposes is path-connectedness . Informally, a set s rn is path-connected if any two points in s can be connected by a path. A set s rn is path-connected1 if, for every pair of points x and y in s, (1) = y. (1) there exists a continuous function : [0, 1] s such that (0) = x, 1as is usual with mathematics, different sources sometimes use different terms. For example, folland uses the terms arcwise connected or pathwise connected in place of path-connected .
