Let f : Rn → R be convex with dom(f) = Ω ⊂ Rn an open set. Show that f is a continuous function on Ω

Q: ove that there does not exist a linear map T: R³ → R³ such that range T = null T (or uivalently…

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Q: Let U be connected open subset of C. Determine all the analytic functions f: U → C such that Im f(z)…

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Q: Let f: [a, b] → R be a bounded function, and let P be a partition of [a, b]. Let P be a refinement…

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Q: Show that if f:R → R is odd and the restriction of fto the interval [0, co) is strictly decreasing,…

A: The odd functions are functions that return their negative inverse when x is replaced with –x. This…

Q: Let f:R→R. If f is defined b then it is Cantindous. foo=2x+1 if X>o if X5o

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Q: Let f: R" x R → R define as follows f(x,y) = a, then f is

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Q: 2. Let f : [a, b] R be a bounded function. Prove that if f is continuous at ro € [a, b), then L(|f])…

A: Here we use the property of continuty.

Q: Let DCR and let f: D→ R be a function. For a set B CR the preimage of B under ƒ is f−¹(B) = {x € D |…

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Q: 3. Let f: UX be a function such that UX. Show that if U is compact, then the range of f defined by…

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Q: Q6. Let f : R? → R² be a function defined by f(t, s) = (1+3t, 2s – t).

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Q: Let f : R" → R be a function. Consider the converse proposition of (6.1) given below: If La(f) is a…

A: Consider the converse proposition of (6.1) given below: if ℒα(f) is convex set for each α∈ℝ, then f…

Q: 3. Assume f: ZR and prove that f is uniformly continuous.

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Q: Prove that if R: R -> R is continuous, theis b2n the image of a bounded set over f is bounded

A: As per the question we have to prove that if any function f : ℝ → ℝ is continuous, then the image of…

Q: Let f,g : R → R be bounded and uniformly continuous. Prove that fg is also bounded and iniformly…

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Q: Prove that if f : R → R is continuous, then the image of a bounded set over f is bounded.

A: Given f:R->R is continuous function  Let A is bounded subset of R  So the limit point set A' is…

Q: Ja (16) Let f e Lla, b]. Show that if g is a bounded measurable function then fa E Cla bl

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Q: Let f: X→Y and g: Y→Z be functions. Prove that if f is an onto function and g∘f is a one-to-one…

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Q: f. Is the domain of f(x.y)= In (6x + 5y“) bounded or unbounded? O Unbounded O Bounded

A: the domain will be Unbounded because

Q: 4. Let A C R" be both connected and sequentially compact. Prove that if F: A –→ R is continuous,…

A: Consider the given information.

Q: (a) Define a function f: N→ N that is one-to-one but not onto. (b) Define a function f: N→ N that is…

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Q: Prove that if f : V → W is differentiable at a point a ∈ V it must be continuous there.

A: I have proved limiting value is equal to functional value

Q: Suppose f: R→→ R is convex. Show that for all x € [a, b]. x-a :) ≤ b = = f(a) + ² = f(b) b a f(x) ≤

A: Suppose   is convex. We have to show that:

Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

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Q: Let f:R→R with the usual topology that maps compact sets to compact sets in R, is the function…

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Q: Give an example (with justification) of a continuous function f : R→R such that for every non-empty…

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Q: Let X, Y and Z be sets. A, B are subsets of Y. f:X → Y is a function. Prove the given statement if…

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Q: Let f: X→Y be a function. Please prove that if f(X-A)=Y-f(A) for any A⊆X, then f is onto.

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Q: Suppose that ci < c2 and that f takes on local maxima at ci and c2. Prove that if f is continuous on…

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Q: Let f: X Y be a function, where X and Y are non-empty sets. If f is onto, prove there exists a…

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Q: Suppose f : N→N and g :N→N , function f is not onto but f+g is onto function  does g onto function…

A: g is not necessary to be onto. See the attachment

Q: Let f be a real-valued function which is bounded on (a, 6), P be a partition of (a, b] and sup{f(r)…

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Q: Let ƒ be a mapping defined by f : x + iy → -Y x Show that f(2122) = f(21)f(22) V 21, 22 E C.

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Q: Let f : (X,d) - (Y, d1) becontinuous fumction, then for any closed set G in X, f(G) is not necessary…

A: False

Q: Suppose ƒ(.) : R" → R is proper. Show that f(-) is convex i and only if epi(f) is a convex subset in…

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Q: Prove that if f: R → R is continuous, then the image of a bounded set over f is bounded.

A: We need to prove that, if  is continuous, then the image of a bounded set over  is bounded.We know…