Preservation of compact sets to form disjoint images

Question

True or False: There is a continuous function $f$ under which the image of $[0,1]$ is $[1,2]\cup[3,4]$

My gut says true since continuous functions preserve compact sets. But I can't seem to form this piecewise function that gives that image. Thanks.

Answer 1

This is false: continuous functions preserve path-connectedness (see here). The source $[0,1]$ is path-connected whereas the target $[1,2]\cup[3,4]$ isn’t. Thus no such $f$ may exist.