(2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies the conclusion of the Intermediate Value Theorem.

Q: Suppose that f: [0,1]–R is a function satisfying the conclusions of the intermediate value theorem.…

A:

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

A: Continuity of a function can be done by many methods as,

Q: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

A: Solution is given below:

Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

Q: Provide a function f, continuous on (0, 1), such that for all r where f'(x) exists, f'(x) > 0 but f…

A:

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

Q: Let f and g be continuous functions on the interval [a,b], such that f(x)=g(x)-1 for all asx<b.…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let f be a real valued function defined on [a, b] . Let f have a local minimum at a point x ∈ (a, b)…

A:

Q: Let f : R → R be a function such that |f (x) – f(y)| 0. Write the e-d definition of continuous…

A:

Q: Consider the function f (2) = (2z+ 1) (z² – 1). - (a) What can be said about the continuity and…

A: The function f(z) = (2z + 1)(z2 - 1)  at the point z = - 1+ 2i is  ? Continious and differentiable.…

Q: D. Prove that f(x) = |x - 6| is continuous at all points c in R.

A: Given function is fx=x−6. We have to prove that the function fx=x−6 is continuous at all point c in…

Q: Suppose f is differentiable on some open interval containing e and f" (c) = 0 then (c, f(c)) is a…

A:

Q: Let fbe a continuous function with domain R such that f(1) = 0.05. Prove that fis strictly positive…

A:

Q: Prove that if f (x) is continuous at x = a and g(y) is continuous at y = b, then F(x,y) = f(x)g(y)…

A:

Q: Suppose that f is continuous at 0, and that f satisfies f(x + y) = f(x)+ f(y) for all real numbers…

A:

Q: 6. Let f: R? → R? be a continuous function. Prove that f ([0,1] × [0,1]) is compact.

A:

Q: If f : R+ → R is a function such that f(x + y) = f(x)f(y) for all x,y ∈ R+. Prove that f is…

A:

Q: 2.) Determine the set of points at which the function is continuous. а. F(x,у) — 1+x²+y² 1-x2-у? ху…

A: Here, to find the set of points at which the function is continous.

Q: Let f be a continuous function defined on E. Is it true that f-'(A) is always measurable if A is…

A: If you mean AA is Lebesgue measurable, then the conclusion is NO. A classic counterexample is…

Q: (b) A function f(x) continuous at 0 and g(x) not continuous at 0 such that f(x)+g(x) is continuous…

A: A function fx is continuous at  x=a, if limx→a+f(x)  = limx→a-f(x) = f(a)

Q: Determine the domain of the function and prove that it is continuous on its domain using Theorems…

A: Consider the given function: fx=xsinx Here, the objective is to determine the domain of the function…

Q: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

Q: If a function f(x) is not defined at x-z, then it is not continuous at interval (a, b) where a > z <…

A: Solution

Q: Let f : R → R with f(x) = 3x² + 2. Prove that f is differentiable at the point –1 %3D

A: f(x) =3x2+2 f'(x) =limh→0f(-1+h) -f(-1)h        =limh→03(-1+h)2+2-3-2h        =limh→03+3h2-6h -3h…

Q: Let g : R → R be a positive function such that g(0) = 1 and g(x+y) = g(x)g(y), ∀x, y ∈ R. Show that…

A:

Q: 3. Let f : R→ R be continuous and let a > 0. Define g : R → R by rx+a g(x) = f(t)dt, x € R. x-a Show…

A:

Q: Can Rolle's Theorem be applied to f(x) = 1/(1-x) on interval [0,3]? If yes, what is c in (0,3) such…

A: here we can't apply the rolles Theorem

Q: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

A:

Q: Consider a surjective function f : A → A Prove that fo f o f is surjective

A:

Q: Find all values of a such that f is continuous on R: |x + 1 if x a

A: For a continuous function LHL ( left hand limt) should be equal to RHL (right hand limit)

Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

A:

Q: True or False A function f has a local maximum at c if there is an open interval I containing c such…

A: We know that, from the definition of local maximum or relative maximum, we say that f has a relative…

Q: Let f be a continuous function on [0, 1]. Suppose that 0 < f(x) < 1 for all x e (0, 1]. Prove that…

A:

Q: Let f(x) = 0 if x∈Q if x and f(x) = 1 if x ∈R\Q. Prove that y=f(x) is not continuous at any point…

A:

Q: Determine the value of c such that the function is continuous on the entire line. f(x) = {* + 3, xs2…

A:

Q: Let f be a continuous function with f(x) > 0 for all æ € R. Define r(z²+1)" g(r) = e" f(t)dt.

A: Let f be a continuous function with fx&gt;0 ∀x∈ℝ. A function g is defined by  gx=∫0x2+1xexftdt…

Q: Let f(x) be a given continuous function on a closed interval [a, b]. Use the Extreme Value Theorem…

A: given fx is a continuous function on a closed interval a,b to prove- use extreme value theorem to…

Q: Let f be a continuous function on [a, b] such that f(r)a" dx = 0 for every n > 0. Prove that f = 0…

A:

Q: 5.Prove that: if f is continuous at x = 0, then xf (x) is differentiable at x = 0.

A: We want to prove the given result.

Q: Let f be a continuous function defined on E. Is it true that f^-1(A) is always measurable if A is…

A: Prove the given statement

Q: 4. Let A := (0, 1] and let f : A → R be defined by f(x) = !. Prove that f is continuous on A.

A:

Q: (c) A function f(x) continuous at 0 and g(x) not continuous at 0 such that f(x)g(x) is continuous at…

A:

Q: Let f : (0, 1) → R given by f(x) = 1/x . (i) Prove that f is continuous. (ii) Prove that f is not…

A:

Q: Suppose f is a differentiable function on an interval II and f(x) 6= 0 for every x in the interval.…

A:

Q: Let A := (0,1] and let f : A → R be defined by f(x) = . Prove that f is continuous on A.

A: Consider the provided question,

Q: 15. Let f. g : R → R be continuous at a point c, and let h(x) := sup (f (x). g(x)} for x e R. Show…

A:

Q: 16. Consider the function f:(-2, 0) → [0, 0) defined by f (x) = vx +2. Prove that f is injective.

A: f(x) here is injective if and only if for all a,b∈(-2, ∞) we have a=b⇔f(a)=f(b)

Q: Prove that f(x) = x 3 - l2x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Given information: The function is f(x)=x3-x. Calculation: We need to find the value of f'(x) and…

Q: Prove that there is no continuous function f : S² → S' such that f(-x) =-f(x) for all x € S².

A: