The discussion revolves around the continuity of the composition of two continuous functions, specifically examining the functions f: D-->R and g: R-->Y, where D, R, and Y are subsets of the plane.
The discussion is active, with participants exploring different definitions of continuity and how they apply to the problem. Some guidance has been offered regarding the use of definitions and the relationship between the functions involved.
There is an emphasis on the need for clarity regarding the definitions of continuity, as multiple equivalent definitions exist. Participants are encouraged to consider the implications of these definitions in the context of the problem.
mikki said:Show that the composition g o f is a continuous transformation.
mikki said:Suppose that f: D-->R and g: R-->Y are two continuous transfromations, where D, R, and Y are subsets of the plane. Show that the composition
g o f is a continuous transformation.
HallsofIvy said:What definition of "continuous" are you using? There are several equivalent ones. The most common is "f is continuous if and only if for every open set U in the range, f-1(U) is an open set in the domain. What is (gof)-1?