4. For a function f:D → R and a point xo in D, define A = {x in D x ≥ xo} and B: {x in D x ≤ xo). Prove that f: D → R is continuous at xo if and only if → R and f: B → R are continuous at xo. f: A =

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Q: 7. Let u be transcendental over F and F< EC F(u). Show that u is algebraic over E.

A: Algebraic field

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A: Step:-1 Mean Value Theorem:-  Let a function f : a, b →ℝ be such that 1. f is continuous on a, b and…

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Q: AP1) Prove that f: E → R; x x² is uniformly continuous if E is bounded. AP2) Prove that f : R R; x x…

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A: We will use of intermediate value theorem to proof this

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Q: Prove that f(x) = x ⋅ |x| is continuous at all points c in R

A: Given the function fx=x·x. We have to prove that fx=x·x is continuous for all points c in ℝ.

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A: Here we will prove that d is a metric function.

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A: Consider the given information.

Q: Suppose that A,B⊂S such that Pr[A|B]=25, Pr[A]=12 and Pr[B′]=12. What is Pr[B|A′]?

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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

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A: Step 1: Step 2: Step 3: Step 4:

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Q: C is the curve y = e® from (2, e²) to (0,1) and F = (x², -y), compute ſ, F-dr.

A: Given that, C is the curve y=ex from 2,e2 to 0,1 and F=x2,-y Then, F=x2,-y=x2,-ex So, F→=x2i^-exj^

Q: Let A = {1, 2, 3} and let f : A→ A, defined by f = (1, 2), (2, 3), (3, 1). Find f, if it exists. %3D

A: