7. Assume that f:XY is a function from one metric space (X,d,) to another(Y,d,). Prove that f is continuous on X if, and only if, f'(int B)cint f"(B) for every subset B of Y.

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Q: As the fun ction Leay) xy js Continuous I+ x*+y? e (-1, 2)

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A: Applying mean value theorem,

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Q: If fx and fy are each continuous in an open region R, is f(x, y) continuous in R? Explain.

A: If fx and fy are each continuous in an open region R, is f(x, y) continuous in R? Explain.

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Q: iii If the discontinuity of ƒ is removable, redefine ƒ so that it's continuous at 0

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Q: 1) Prove that for any space, X, with any topology, T, that: (i) CI(X) = X and (ii) Int(X) = X.

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Q: Suppese that X, X and that f is continuous. Show that f(X,) 1(X). Show w that ETLJY

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Q: metric

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Q: Suppose that f is integrable on [a,b], and that g(x) is defined for all x E[a,b] and ,exists and…

A: Given: To explain the given statement as follows, To prove that    exists and equals .

Q: Let f : AC X → Y be a continuous function between metric spaces. If A is open then so is f(A).

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Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

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Q: Let f : [a, b] → R be a differentiable function with f′ bounded. Show that f is is uniformly…

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Q: Given, we have a finite purely non-atomic measure space (X, F, µ), prove that µ(F) = [0 µ(X)].

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