Q. Consider the measure space (R, M(R), µ), if x € Q 1 -2 let k(x) = { , h(x) = -2,then %3D ¢ Q if x a.k(x) = h(x) a.e. and f is continuous a.e. b. k(x) # h(x) a.e. and f is not continuous a.e. c.k(x) + h(x) a.e. and f is continuous a.e. d. k(x) = h(x) a.e. and f is not continuous a.e. e. None of these.

Q: 'he function f: E → R is uniformly continuous if

A: We have to find the answer using the definition of uniform continuity.

Q: Q. Let g(x) : [2, 3] -+ R be a continuous mapping. 3 And / (9(x) + 2 – 1)'dx = 0, then - a.g(x) = 0.…

A: Given function g(x) is continuous on the interval [2,3]

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]-->R is a continuous function then f is integrable over [a,b].

Q: 21. Let f(x) = 2x, x = [0, 1]. Take P = {0, 1, 1, 1, 1, 1} and set x₁=1, x₂ = 16, x3 = 3, x4={, x3 =…

A:

Q: 1. Show that the following functions f : R R are continuous. You should argue carefully from the…

A:

Q: Problem (3) Show that if a continuous function f is holomorphic outside a line in C then it is…

A:

Q: The set of functions S = span{e™" | n e Z} separates [0,1]). True False

A: S=span{enx |n∈Z}

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

A:

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: The function f is a composite of two continuous functions: y = cos x and y = x 2, so f is continuous…

A: The function fx is a composition of functions y=cosx and y=x2and it is a continuous function.

Q: Answer "True" or "False" for the following: Let f(x) be a continuous function on [a, b]. Suppose…

A: Definition of the antiderivatives helps to validate the given statement. A function F(x) is said to…

Q: Prove or disprove: If f is continuous, then S s) dx = [r(1-x) dx.

A:

Q: 4

A:

Q: If f is continuous everywhere, then af(bx) dæ < af(bx) dx, where a and b are real numbers. -3 O True…

A:

Q: Prove thait if y=f(x) f(x)=0 when x is a rotional number, then f(x)=0 for all x € R. is continuous…

A:

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)Df(x)+fy) vx, yER. Show that…

A:

Q: 11. Let f(x, y) = x² - y, g(t) = (Int) In(1-t), and h(x, y) = g(f(x, y)). Find the set on which h is…

A: Given that  f (x, y) = x² - y g(t) = (In t) In(1 - t) We have to find the Composite Function h(x, y)…

Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

A:

Q: H IS A CONTINUOUS FUNCTION HCX) >-1 FOR EVFRY X FOR EVER4 IW [a,b J. Then dx

A: Given that H(x) &gt; -1 for all x in [a,b] If -1 &lt; H(x) &lt; 0 for all x , then the integration…

Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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: 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: {4, X, {m}, {n}}. [0, 0) by Q. Let X = {m, n}, and = %3D Define the function um : 2 if me A then…

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.