1. (a) Show that any open set U in R is a union of countably many disjoint open intervals, one or two of which may be unbounded (the line or half-lines). Hint: For each x ∈ U , find a largest open interval containing x and included in U.
(b) Let F be a closed set in R whose complement is the union of disjoint open intervals (an, bn ) from part (a). Let f be a continuous real-valued function on F . Extend f to be linear on [an, bn ], or if an or bn is infinite, extend f to be constant on a half-line. Show that the resulting f is continuous on R. Note: Be aware that subsequences of the an may converge.