2. Construct a sequence of unbounded functions In : [0,1] → R that converges pointwise to a bounded function g : [0, 1] → R.
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![2. Construct a sequence of unbounded functions In : [0,1] → R that converges
pointwise to a bounded function g : [0, 1] → R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F23771f71-579b-4bb4-a754-662cbf5420cc%2Faczwp5d_processed.jpeg&w=3840&q=75)
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- Suppose that a sequence of differentiable functions {fn} converges pointwiseto a function f on an interval [a,b], and the sequence {f′n}converges uniformlyto a function g on [a,b]. Then show that f is differentiable and f′(x) = g(x)on [a,b].2. Suppose for every n ≥ 1 that {f} are twice differentiable functions on [a, b] such that both {f} and {f} converge uniformly on [a, b]. Prove or disprove that {f} converges uniformly on [a, b].4. Prove that given a continuous function f on [a, b], there exists a sequence (Pn) of polynomials such that pn →f uniformly on [a, b], and for each n, pn(a) = f(a) and pn (b) = f(b). Hint: Select a sequence (an) of polynomials such that qnf uniformly on [a, b]. For each n, let sn be the function on R whose graph is the straight line passing through the points (a, f(a)- qn(a)) and (b, f(b)- qn (b)). Set pn = 9n + Sn.
- Prove that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is uniformly convergent on R.2. Suppose that f is an increasing function on the interval [a, b]. Let F(t) | f(x)dr. a.) Argue that if a < x* < b then f(a) < f(x*)< f(b). b.) Prove that F is continuous over [a, b].Show that if each fn is bounded on A and (fn) converges uniformly to f on A, then f is bounded on A.
- If fn is a uniformly equicontinuous sequence of functions on a compact interval and fn → ƒ pointwise, prove that fn → f uni- formly. (You should not assume that f is continuous, although this is a consequence of the result.)Give an example of a bounded function that is defined on R, continuous at each point except the points 1 and-1, and has neither a removable discontinuity nor a jump discontinuity at 1 and-1.A balanced ternary string of length n is a function f: [n] → {-1,0, 1}. The weight of such a string is the sum f(1) + f(2) + … + f(n). Show that the number of balanced ternary strings with weight 0 is 2k k20 where ak = G) if 2k < n, and 0 otherwise.
- 2n 8n Show that f, (x) = x" – x" is unfiormly converges or not by using uniform normLet (fn) be a sequence of Lebesgue measurable functions on [a, b] such that fn → funiformly on [a, b]. Show thatZ baf = limn→∞ Z bafn.Let (Jn)neN be a sequence of functions f: R → R satisfying |fn+1(c) – fn(c)| 2 and neN , and ae(0, 1). Prove that (Sn)n€N is a pointwise convergent sequence of functions at x = c.
