Let F: [0, 1] [0, o0] be a measurable function. Assume that lim sup,x " dr < o. Show that f(r)S1 for almost all r E (0, 1]. Hint: If a >1 then the sequence of powers a, a,... goes to infinity.

Q: Determine whether f(t) is continuous, piecowise continuous, or neither on (0, 40, Ost<3 (1)= t+13,…

A: Explained below

Q: x(x²+3) Q7. Consider the function f : [0, 1] → R, → √√4+24 Explain why (a) f continuous, (b) f is…

A:

Q: Suppose ƒ is continuous on [1, 5] and 2 ≤ƒ(x) ≤ 3, for all x in[1, 5]. Find lower and upper bounds…

A: If f:[a,b]--&gt;R is a continuous function then f is integrable over [a,b].

Q: 12. For each n € N, define A, = [0, 1+] and B₁ = [0,1]. - =1 (a) Prove that U1A,, is an interval,…

A:

Q: The function f is continuous on [0, 4]. f(0)= -2 and f(4) = 5. Must there exist a number C between 0…

A: Given,         The function f is continuous on 0,4. f0=-2 and f4= 5.         There exist a number c…

Q: 7. Suppose f is a continuous function on (a, b) with b> a. Stat f (recall this theorem was taught in…

A:

Q: 2. Suppose that f is an increasing function on the interval [a, b]. Let F(t) = %3D a.) Argue that if…

A: suppose that f is an increasing function on the interva a,b let Ft=∫0tfxdx (a) if a&lt;x*&lt;b then…

Q: Find the dynamic process equations for the following X₁ = 2x₁ 1 X ₂ = x₁ te - 2X2 -34 - time -…

A:

Q: Q4) let f be non-negative continuous function of [a, o]& k> 0 is constant. IF t f(t) a holds, then…

A: Given f is a non-negative continuous function on [a,∞) and k&gt;0 is a constant. Suppose,…

Q: (5) Consider sets A and B with |A| = sand |B| = t, s≥ t≥1. (a) How many onto functions are there…

A: Sets, A and B with A=s and B=t, s≥t≥1.

Q: Suppose that f: A → R is continuous at a number a E A with f(a) > 0. Show that there exist numbers L…

A:

Q: Let f(x) = 36 – 5z – z?. (A) Use interval notation to indicate where f(z) is increasing. Note: Use…

A: Answers are provided below.

Q: Let 1 f(x) = ,* – 8, – 4 < x < 8 1 The domain of f¯'is the interval [A, B] where A and B = Check…

A:

Q: fn. (1,00) Evaluate lim

A:

Q: 1 If the function f is continuous on [a,b], then there exists at least one number C in [a,b] such…

A: If the function f is continuous on [a, b], then there exists at least one number C in [a, b] such…

Q: 4. Let f and g be functions defined on an open interval I containing the point c. Prove that if f…

A:

Q: Un which of the following intervals is 1 f(x) = continuous? Vx-2 (2, + ) (-a, +) [1,2)

A:

Q: . If f : (0, 3) → R is continuous, must its range be an open (possibly infinite) interval?

A: See the detailed solution below.

Q: 6. Let S = {-1,0, 2, 4, 7}. Find f(S) (image of S under f) where f is from R to R for (a) f(a) = 1…

A: The

Q: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

A:

Q: Where is f(x)= -5 -6z+5 discontinuous?

A: The given function is fx=x-5x2+6x-5. We have to find the point where the given function is…

Q: x+2, x0 a. (-0, 0) b. (-0, 0] [-1, 1) d. (0, 1] e. [0, +0) с.

A: A function f(x) is said to be discontinuous at x=a if limx→a+f(x)=limx→a-f(x)=f(a)

Q: Suppose that f and g are both continuous functions. Also f, g € [a, b] and that f= fg. Prove that 3c…

A:

Q: 4. If continuous function f:[0,1] U (2,3) → R is one-to-one, is f also continuous? 1

A:

Q: If that f is continuous on a sxsb. Let F(x) be any of f(x). Then, f(x)dx = F(b) - F(a). This theorem…

A: Recall: Mean value theorem: If f(x) is continuous over an interval [a,b], then there is atleast one…

Q: Which of the following functions is continuous on the interval (-0, +)? …* f(x) = Vx² – 1 (A f(x) =…

A:

Q: Let the function ƒ : [0, 1] →R be defined by { if x = ! (n = 1, 2, 3, · · · ) f(x) = n 1– x2…

A: Given :  f : [0, 1] →ℝ  defined by ;              fx = ex2,if  x=1n    (n=1, 2, 3, ...…

Q: A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1…

A:

Q: (2) Show that C(0, 1] C L'(0,1]. Compare |flsup and fl. Justify your answer. Generalize this to the…

A:

Q: -2 x<-1 Q1)a) For what values of a and b is g(x) = |ax - b - 1<x< 1, 3 x21 Continuous at x=1

A: From  the given problem:   gx=-2x≤-1ax-b-1≤x&lt;13x≥1     As we know that a function f(x) is…

Q: Let n > 2 be a natural number. Prove that there is no continuous function q : C → C such that…

A:

Q: Determine whether -2, if z < -2 *, if - 22122 |8- z), if 2<z <3 |4 -r, if z 23 f(a) = is continuous…

A: givenf(a)=-2, ifx&lt;-2x5 , if-2≤x≤28-x , if2&lt;x&lt;34-x2 , ifx≥3one limit of this piecewise…

Q: c) Consider a function f defined on the real numbers as { 0ifais irrational. S(r) = 1/n ifr= m/n…

A: To show that the function f is continuous at all irrational numbers.