(a) Let ƒ : [a, b] → R be continuous with f(a) < 0 and f(b) > 0, and S := {x = [a, b] | f(x) < 0}. If c is the supremum of S, prove that f(c) = 0. (b) Suppose that ƒ : (0,1) → R is uniformly continuous. Prove that limx→o+ f(x) exists.

Q: The function f is defined by f (x) = x² sin. The application of the squeeze theorem guarantees that…

A:

Q: Let f and g be real-valued functions defined on (b, ∞). Suppose that limx->∞f(x)=L and…

A:

Q: Let f:R→R satisfy f(x+y) =f(x) +f(y), ∀x,y∈R. If limx→0f(x) exists, prove that f is continuous on R

A:

Q: Consider a function f : D → R, where D C R. Suppose the function satisfies the property f(xy) = f(x)…

A: A function f:D→R, where D⊂R f(xy)=f(x)+f(y) for all x, y in D f(x) is not zero for all x in D.

Q: Suppose that f is twice differentiable at a. (a) Prove that there exist a 0ħ € (0, 1) such that f(x…

A: 3) a)  Suppose  that f   is  twice differentiable at a.We  have  to  show  that  there  exists a…

Q: Consider the following proposed variation of Rolle's theorem. Let f: (a, b) → R be a differentiable…

A: We have defined a special real valued differential on (0,1) .

Q: If f is continuous and ƒ(5) = 2 and ƒ(4) = 3 then lim ƒ(4x² – 11) = x→2 True False

A:

Q: (Group B) Suppose f: [0, 1] → R is a continuous function such that f(0) = f(1). Prove [0,1] such…

A: Let us consider the function f:0, 1→ℝ. Here f is a continuous function such that f0=f1. To prove…

Q: If a function f : R” → R is differentiable at Xo, prove that ƒ is continuous there. Let ƒ : R" → R.…

A: Given the function f:ℝn→ℝ is differentiable at X0. To prove f is continuous. Given f:ℝn→ℝ and there…

Q: (a) Letƒ be differentiable on (a, b). Prove that if f'(x) #0 for each x e (a, b), then ƒ has at most…

A:

Q: True or false: If f: R\ {0} → R is uniformly continuous, then lim f(x) 2-0 exists. True False

A: f: ℝ\0→ℝ is a uniform continuous function , We have to verify whether limx→0fx exist or not

Q: Let f be a function defined on an open interval I containing a. Prove that f'(a) exists and is equal…

A:

Q: a. If f is continuous, then ſ f(x) dx = lim ſ´, f(x) dx. t-00 b. If f(x) < g(x) and g(x) dx…

A: a.) If f is continous then the given condition on the integral is always true. b.) This is a false…

Q: Let f be a continuous function on R such that limr→∞ f (x) = L* and lim,-0 f(x) = L-. Assume L+ > L¯…

A:

Q: (3) Suppose that f: [0, 1] → R is a real function. with f([0, 1]) ≤ [0, 1]. Show that there exists…

A: Sol:- Let g(x) = x - f(x). Then g(x) is also a function from [0, 1] to R. Notice that g(0) = 0 -…

Q: Let f, g: R → R be continuous at a point c, and let h(x) = sup{f(x), g(x)} for x € R. Show that h(x)…

A:

Q: Prove that if S(c +h) – f(c) lim h exists, then f is continuous at c.

A:

Q: Let g:BX C- C(with BØ) be the function given by g (x, y) = y. (a) Prove that g is surjective. (b)…

A:

Q: A jungle has 400 monkeys, and it is estimated that the maximum population of the monkeys in the…

A:

Q: 4. (i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is…

A: According to mean value theorem , If f(x)f(x) is continuous over an interval [a,b],[a,b], then there…

Q: prove

A: To show that for every c in [−1,1] there exists a sequence of points xn​ on (0,∞) such that lim⁡n→∞…

Q: Find an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) → 0…

A: To give an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) →…

Q: Let I (a, b) be a bounded open interval and ƒ : I → be a monotone increasing function on I. (i) If f…

A:

Q: Theorem 4.2.5 (Cauchy's mean value theorem). Let f : [a,b] –→R and o: [a,b] → R be continuous…

A:

Q: f(x) = √x  is Lipschitz continuous (b) Prove that for f: [1, infinity) defined by f(x) = √x  is

A: (a). According to the question, to show that f(x) = √x is Lipschitz continuous on [1, infinity), we…