1. One can construct Riemann integrable functions on [0,1] that have a dense set of discontinuities as follows. (a) Let f(x) = 0 when x < 0, and f(x) = 1 if x > 0. Choose a countable dense sequence {rn} in [0, 1]. Then, show that the function F(z) =Ef(x – r.) n=1 is integrable and has discontinuities at all points of the sequence {r,}. [Hint: F is monotonic and bounded.] (b) Consider next F(x) = 3-"g(x – rn), n=1 where g(r) = sin 1/x when r + 0, and g(0) = 0. Then F is integrable, discontinuous at each r = rn, and fails to be monotonic in any subinterval of (0, 1). [Hint: Use the fact that 3-k > En>k3-".]

Q: 10. Let f(0) = 0 and f(x) = 1/x for all x € (0, 1]. Show that f is not integrable on [0, 1] by…

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A: Given that A shopper buys cat food in bags of 3 lbs. Her cat eats 3/4 lb each week

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Q: (a) Define a sequence of functions on R by

A: Given: (a) fnx=1if x=1, 12, 13, . . . ,1n0otherwise f is a pointwise function of limit fn. To Find:…

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Q: : . Let fn [a, b] → R be a sequence of Riemann integrable functions such that fn → f uniformly for…

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Q: 2. Show that the function, defined by f(x) = is NOT uniformly continuous on its x2 %3D domain D(f) =…

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Q: {fn} integrable on u fn => f and [a,b] [a, b]. Prove f integrable on on Sa f = lim Sha fn n 0 [a,b]

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Q: Suppose that f is continuous on that 9< f(4) < 21. [0, 4], f(0) = 1, and 2 < f'(x) < 5 for all a in…

A: Here in this question we have a function which is both continuous and differentiable in the given…

Q: 9 Prove that if m(x) is differentiable on (-∞, 0) and its derivative m'(x) is bounded then m is…

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Q: 1) It shows that the function f(x) = 1/x  is integrable in the closed interval [1, 2]. This must be…

A: As you asked more than one questions I only solve the first one.

Q: Compute the left and right Riemann sums L4 and R4, respectively for f(x) = √4x² on [-2, 2] and…

A: We can find Left and Right Riemann sums .

Q: 1. Let {gn} be a sequence of integrable functions which converges a.e. to an integrable function g.…

A: Solution: Given gn is a sequence of integrable functions which converges a.e. to g. Also, fn is a…

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A: We will find out the required value.

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Q: 6. Let f(x) = x2/3. (a) Then fis uniformly continuous on [-1,1]. Why? (b) Show that there does not…

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A: Note: We are entitled to solve only one question at a time and up to 3 sub-parts only. As the…

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A: The theorem is done .

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A: Given that The pointwise limit is,

Q: 3. Let f be integrable on R. For each n, let gn be the function defined by In(x) = f(x - 1/n). Prove…

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Q: 22. Let fn R→→ R be a sequence of functions such that f'(x), f"(x) are defined for all x € R.…

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Q: Suppose f is a continuous function defined on (-e, e) for some e > 0 and assume that f(0) = 0 and…

A: Since function f(x) is continuous at x=0. Also given that limx→0f(x)x=C

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Q: Isg continuous on (0,1]? * if x 4.

A: Solution:Given  g(x)=xif x≤03if 0&lt;x≤13-x2if 1&lt;x≤4x-3if x&gt;4To find:g is continuous at (0,1]

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Q: Suppose f(z) is defined on A = {z Re(2) >0} such that f(z) is differentiable at each point in A and…

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Q: Let h: [0,1] R be a continuous function with h(1) by fn(x) = x"h(x) for x e [0,1]. Show that {fn}…

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Q: Let (hn), (tn) be sequences of bounded functions on A that converges uni formly Sh, tn) converges…

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Q: Let f(x) be a continuous function on [0, 1] and consider the limit 1 Xj +... f + In · drn. lim dx1…

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Q: et f : [0, 1] → R be a continuous function on the closed interval [0, 1] and differentiable on the…

A: Given: To show that the given statement as follows, g(x) = f(x)x.