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mikki
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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.
g o f is a continuous transformation.
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?
The composition of continuous maps is the process of combining two or more continuous functions to create a new continuous function. It is denoted as f ∘ g, where f and g are both continuous functions.
The composition of continuous maps is important because it allows us to create more complex and meaningful functions by combining simpler functions. It also plays a crucial role in many mathematical proofs and applications.
To prove that the composition of continuous maps is continuous, we need to show that for any two continuous functions f and g, the composition function f ∘ g is also continuous. This can be done using the epsilon-delta definition of continuity or by using the intermediate value theorem.
No, the composition of discontinuous maps cannot be continuous. In order for the composition of two functions to be continuous, both functions must be continuous. If even one of the functions is discontinuous, the composition will also be discontinuous.
Yes, there are some cases where the composition of continuous maps may not be continuous. For example, if the domain of the first function does not overlap with the range of the second function, the composition may not be defined or continuous. Additionally, if the functions have a removable or jump discontinuity at the point of composition, the resulting composition may still be continuous.