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: 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 =…

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

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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.

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Q: 4

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

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

A: Given that H(x) > -1 for all x in [a,b] If -1 < H(x) < 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).…

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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…

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.