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Introduction to Real Analysis 4th edition Robert G. Bartle, Donald R. Sherbert - Solutions

Let a
Let Î´Ê¹ and Î´Ê¹Ê¹ be as in the preceding exercise and let Î´* be defined by
Let d be a gauge on I: = (a, b) and suppose that I does not have a d-fine partition.(a) Let c : = (a + b). Show that at least one of the intervals [a, c] and [c, b] does not have a Î´-fine partition.(b) Construct a nested sequence (In) of subintervals with the length of In equal to (b - a)/2n
Let I := (a, b) and let f : I → R be a (not necessarily continuous) function. We say that f is ''locally bounded'' at c ∈ I if there exists δ(c) > 0 such that f is bounded on I ∩(c - δ(c), c + δ(c). Prove that if f is locally bounded at every point of I, then f is bounded on I.
Let I := [a, b] and let f : I → R be continuous on I. If f has an absolute maximum [respectively, minimum] at an interior point c of I, show that f is not injective on I
Let f : |0, 1 → R be a continuous function that does not take on any of its values twice and with f(0) < f(1). Show that f is strictly increasing on [0, 1].
Let h : |0, 1| → R be a function that takes on each of its values exactly twice. Show that h cannot be continuous at every point. [If c1 < c2 are the points where h attains its supremum, show that c1 = 0, c2 = 1. Now examine the points where h attains its infimum.]
Let x ∈ R, x > 0. Show that if m, p ∈ Z, n, q ∈ N, and mq = np, then (x1/n)m = (x1/q)p
If x ∈ R, x > 0, and if r, s ∈ Q, show that xrxs = xrþs = xsxr and (xr)s = xrs = (xs)r.
Show that if I := [a, b] and f : I → R is increasing on I, then f is continuous at a if and only if f (a) = inf{ f(x) : x ∈ (a, b]}.
Let I ⊂ R be an interval and let f : I → R be increasing on I. Suppose that c ∈ I is not an endpoint of I. Show that f is continuous at c if and only if there exists a sequence (xn) in I such that xn < c for n = 1, 3, 5, . . . ; xn > c for n = 2, 4, 6, . . . ; and such that c = lim(xn) and
Let I ⊂ R be an interval and let f : I → R be increasing on I. If c is not an endpoint of I, show that the jump jf (c) of f at c is given by inf{f(y) - f(x) : x < c < y, x, y ∈ I}
Let I := [0, 1] and let f : I → R be defined by f (x) := x for x rational, and f(x) := 1 - x for x irrational. Show that f is injective on I and that f (f(x)) = x for all x ∈ I. (Hence f is its own inverse function!) Show that f is continuous only at the point x = 1/2.
Use the definition to find the derivative of each of the following functions: (a) f(x) := x3 for x ∈ R. (b) g(x) := 1/x for x ∈ R; x ≠ 0. (c) h(x) = √x for x > 0. (d) k(x) := 1/√x for x > 0.
Let g : R → R be defined by g(x) := x2sin (1/x2) for x ≠ 0, and g(0) := 0. Show that g is differentiable for all x ∈ R. Also show that the derivative gʹ is not bounded on the interval (-1, 1).
Assume that there exists a function L : (0; ∞) → R such that Lʹ (x) = 1/x for x > 0. Calculate the derivatives of the following functions: (a) f(x) := L(2x + 3) for x > 0. (b) g(x) := (L(x2))3 for x > 0; (c) h(x) := L(ax) for a > 0, x > 0, (d) k(x) := L(L(x)) when L(x) > 0, x > 0.
Let f : I → R be differentiable at c ∈ I. Establish the Straddle Lemma: Given ε > 0 there exists δ(e) > 0 such that if u, v ∈ I satisfy c - δ(e) < u < c < v < c + δ(e), then we have f(v) - f(u) - (v - u) fʹ(c)| < e(v - u). [The δ(e) is given by Definition 6.1.1. Subtract and add the
Prove Theorem 6.1.3(a), (b).
Differentiate and simplify: (a) f(x) := x/1 + x2, (b) g(x) := √5 - 2x + x2, (c) h(x) := (sin xk)m for m, k ∈ N. (d) k(x) := tan (x2) for |x| < √π/2.
Suppose that f : R → R is differentiable at c and that f(c) = 0. Show that g(x) := |f(x)| is differentiable at c if and only if fʹ(c) = 0.
Determine where each of the following functions from R to R is differentiable and find the derivative: (a) f(x) := |x| + |x + 1|. (b) g(x) : = 2x + |x|. (c) h(x) := x|x|. (d) k(x) := |sin x|.
For each of the following functions on R to R, find points of relative extrema, the intervals on which the function is increasing, and those on which it is decreasing: (a) f(x) := x2 - 3x + 5. (b) g(x) := 3x - 4x2; (c) h(x) := x3 - 3x - 4. (d) k(x) := x4 + 2x2 - 4:
Let g : R → R be defined by g(x) := x + 2x2 sin(1/x) for x ≠ 0 and g(0) := 0. Show that gʹ(0) = 1, but in every neighborhood of 0 the derivative gʹ(x) takes on both positive and negative values. Thus g is not monotonic in any neighborhood of 0.
If h(x) := 0 for x < 0 and h(x) := 1 for x > 0, prove there does not exist a function f : R → R such that fʹ(x) = h(x) for all x ∈ R. Give examples of two functions, not differing by a constant, whose derivatives equal h(x) for all x ≠ 0.
Let f : (0, ˆž) †’ R be differentiable on (0, 1) and assume that fÊ¹(x) †’ b as x †’ 1.(b) Show that if f(x) †’ a as x †’ ˆž, then b = 0.
Let I := (a, b) and let f : I †’ R be differentiable at c ˆˆ I. Show that for every Îµ > 0 there exists Î´ > 0 such that if 0
A differentiable function f : I †’ R is said to be uniformly differentiable on I := (a, b) if for every Îµ > 0 there exists Î´ > 0 such that if 0Show that if f is uniformly differentiable on I, then fÊ¹ is continuous on I.
Find the points of relative extrema, the intervals on which the following functions are increasing, and those on which they are decreasing: (a) f(x) := x + 1/x for x ≠ 0. (b) g(x) := x/(x2 + 1) for x ∈ R; (c) h(x) := √x - 2√x + 2 for x > 0. (d) k(x) := 2x + 1/x2 for x ≠ 0.
Find the points of relative extrema of the following functions on the specified domain: (a) f(x) := x2 - 1 for - 4 < x < 4. (b) g(x) := 1 - (x - 1)2/3 for 0 < x < 2; (c) h(x) := x|x2 - 12| for - 2 < x < 3. (d) k(x) := x(x - 8)1/3 for 0 < x < 9:
Let f : (a, b) †’ R be continuous on [a, b] and differentiable in (a, b). Show that if then fÊ¹(a) exists and equals A. [Use the definition of fÊ¹(a) and the Mean Value Theorem.]
Let f : R → R be defined by f(x) := 2x4 + x4 sin(1/x) for x ≠ 0 and f(0) := 0. Show that f has an absolute minimum at x ¼ 0, but that its derivative has both positive and negative values in every neighborhood of 0.
Evaluate the following limits:(a)(b)(c)(d)
Evaluate the following limits:(a)(b) (c) (d)
Let f be differentiable on (0, 1) and suppose that Show that [f(x) = exf(x)/ex.]
In addition to the suppositions of the preceding exercise, let g(x) > 0 for x ˆˆ (a, b), x ‰ c. If A > 0 and B = 0, prove that we must have If A
Let f(x) := x2sin (1/x) for 0 0 for x ‰ 0. Show that does not exist.
Evaluate the following limits.(a)(b)
Evaluate the following limits, where the domain of the quotient is as indicated.(a)(b) (c) (d)
Evaluate the following limits:(a)(b) (c) (d)
Evaluate the following limits:(a)(b) (c) (d)
Let h(x) := e-1/x2 for x ‰ 0 and h(0) := 0. Show that h(n)(0) = 0 for all n ˆˆ N. Conclude that the remainder term in Taylor's Theorem for x0 = 0 does not converge to zero as n †’ ˆž for x ‰ 0. [By L'Hospital's Rule, 0 for any k ˆˆ N.
Calculate e correct to seven decimal places.
Determine whether or not x = 0 is a point of relative extrema of the following functions: (a) f(x) := x3 + 2. (b) g(x) := sin x - x. (c) h(x) := sin x + 1/6 x3. (d) k(x) := cos x - 1 + 1/2 x2:
Let f be continuous on [a, b] and assume the second derivative fʹʹ exists on (a, b). Suppose that the graph of f and the line segment joining the points (a, f(a)) and (b, f(b)) intersect at a point (x0; f(x0)) where a < x0 < b. Show that there exists a point c ∈ (a, b) such that fʹʹ(c) = 0.
If I := (0, 4), calculate the norms of the following partitions: (a) P1 := (0, 1, 2, 4). (b) P2 := (0, 2, 3, 4). (c) P3 := (0, 1, 1.5, 2, 3.4, 4). (d) P4 := (0, .5, 2.5, 3.5, 4)
Consider the Dirichlet function, introduced in Example 5.1.6(g), defined by f(x) := 1 for x ∈(0, 1) rational and f(x) := 0 for x ∈ (0, 1) irrational. Use the preceding exercise to show that f is not Riemann integrable on [0, 1].
Suppose that c 0 for x ˆˆ [c, d] and Ï†(x) = 0 elsewhere in [a, b], prove that Ï† ˆˆ R[a, b] and that ˆ«Rb Ï† = a(d - c). [Given Îµ > 0 let Î´Îµ := Îµ/4a and show that if a(d - c + 2Î´Îµ).]
Let 0(a) Show that qi satisfies 0 (b) Show that Q(qi) (xi - xi-1) = 1/3 (x3i - x3i-1.(d) Use the argument in Example 7.1.4(c) to show that Q ˆˆ R[a, b] and
If f ∈ R[a, b] and c ∈ R, we define g on [a + c, b + c] by g(y) := f(y - c). Prove that g ∈ R[a + c, b + c] and that ∫b+ca+c g = ∫ba f . The function g is called the c-translate of f.
If f (x) := x2 for x ˆˆ (0, 4), calculate the following Riemann sums, where has the same partition points as in Exercise 1, and the tags are selected as indicated.(a) with the tags at the left endpoints of the subintervals.(b) with the tags at the right endpoints of the
Show that f : [a, b] †’ R is Riemann integrable on [a, b] if and only if there exists L ˆˆ R such that for every Îµ > 0 there exists Î´Îµ > 0 such that if is any tagged partition with norm
Let :{(Ii , ti)}ni=1 be a tagged partition of [a, b] and let c1 (a) If u belongs to a subinterval Ii whose tag satisfies c1 .(b) If v ˆˆ [a, b] and satisfies c1 + , then the tag ti of any subinterval Ii that contains v satisfies ti ˆˆ [c1, c2].
(a) Let f(x) := 2 if 0 < x < 1 and f(x) := 1 if 1 < x < 2. Show that f ∈ R[0, 2] and evaluate its integral.(b) Let h(x) := ∈ if 0 < x < 1, h(1) := 3 and h(x) := 1 if 1 < x < 2. Show that h ∈ R[0, 2] and evaluate its integral.
If f ˆˆ R[a, b] and if is any sequence of tagged partitions of [a, b] such that prove that
Let f : [a, b] †’ R. Show that f ˆ‰ R[a, b] if and only if there exists Îµ0 > 0 such that for every n ˆˆ N there exist tagged partitions with such that
If f and g are continuous on [a, b] and if ∫ba f = ∫ba g, prove that there exists c ∈ [a, b] such that f(c) = g(c).
If f is bounded by M on [a, b] and if the restriction of f to every interval [c, b] where c ∈ [a, b] is Riemann integrable, show that f ∈ R[a, b] and that ∫bc f → ∫ba f as c → a+. [Let ac(x) := -M and vc(x) := M for x ∈ [a, c] and ac(x) := ωc(x) := f(x) for x ∈ [c, b]. Apply the
If f is bounded and there is a finite set E such that f is continuous at every point of [a, b]\E, show that f ∈ R[a, b].
If f is continuous on [a, b], a < b, show that there exists c ∈ [a, b] such that we have ∫ba f = f(c)(b - a). This result is sometimes called the Mean Value Theorem for Integrals.
If f and g are continuous on [a, b] and g(x) > 0 for all x ∈ [a, b], show that there exists c ∈ [a, b] such that ∫ba fg = f(c) ∫ba g. Show that this conclusion fails if we do not have g(x) > 0. (This result is an extension of the preceding exercise.)
Let f be continuous on [a, b], let f(x) > 0 for x ∈ [a, b], and let Mn :(∫ba fn)1/n. Show that lim (Mn) = sup{f(x) : x ∈ [a, b]g.
Suppose that a > 0 and that f ∈ R[-a, a].(a) If f is even (that is, if f (-x) = f(x) for all x ∈ [0, a], show that ∫a-a f = 2 ∫a0 f.(b) If f is odd (that is, if f (-x) = -f(x) for all x ∈ [0, a], show that ∫a-a f = 0.
Suppose that f is continuous on [a, b], that f(x) > 0 for all x ∈ [a, b] and that ∫ba f = 0. Prove that f(x) = 0 for all x ∈ [a, b].
Extend the proof of the Fundamental Theorem 7.3.1 to the case of an arbitrary finite set E.
Let f : [a, b] → R be continuous on [a, b] and let v : [c, d] → R be differentiable on [c, d] with v ([c, d)] ⊂ [a, b]. If we define G(x) := ∫v(x)a f , show that Gʹx = f(v(x)) ∙ vʹ(x) for all x ∈ [c, d].
Let f : [0, 3] → R be defined by f(x) := x for 0 < x < 1, f(x) := 1 for 1 < x < 2 and f(x) := x for 2 < x < 3. Obtain formulas for F(x) := ∫x0 f and sketch the graphs of f and F. Where is F differentiable? Evaluate Fʹ(x) at all such points.
The function g is defined on [0, 3] by g(x) := -1 if 0 < x < 2 and g(x) := 1 if 2 < x < 3. Find the indefinite integral G(x) = ∫x0 g for 0 < x < 3, and sketch the graphs of g and G. Does Gʹ (x) = g(x) for all x in [0, 3]?
Show there does not exist a continuously differentiable function f on [0, 2] such that f(0) = -1, f(2) = 4, and fʹ(x) < 2 for 0 < x < 2. (Apply the Fundamental Theorem.)
Explain why Theorem 7.3.8 and/or Exercise 7.3.17 cannot be applied to evaluate the following integrals, using the indicated substitution.(a)(b) (c) (d)
Let f ; g ∈ R[a, b].(a) If t ∈ R, show that ∫ba (tf ± g)2 > 0.(b) Use (a) to show that 2|∫ba f g| < t ∫ba f2 + (1/t) ∫ba g2 for t > 0.(c) If ∫ba f2 = 0, show that ∫ba f g = 0.(d) Now prove that |∫ba f g|2 < (∫ba | f g|)2 < (∫ba f2) ∙ (∫ba g2). This
We have seen in Example 7.1.7 that Thomae's function is inR[0, 1] with integral equal to 0. Can the Fundamental Theorem 7.3.1 be used to obtain this conclusion? Explain your answer.
Let F(x) be defined for x > 0 by F(x) := (n - 1)x - (n - 1)n/2 for x ∈ [n - 1, n], n ∈ N. Show that F is continuous and evaluate Fʹ(x) at points where this derivative exists. Use this result to evaluate ∫ba [[x]]dx for 0 < a < b, where [x]] denotes the greatest integer in x, as
Let f (x) := |x| for -1 < x < 2. Calculate L(f; P) and U(f; P) for the following partitions: (a) P1 := (-1, 0, 1, 2). (b) P2 := (-1, -1/2, 0, 1/2, 1, 3/2, 2).
If f is a bounded function on [a, b] such that f(x) = 0 except for x in {c1, c2, . . . , cn} in [a, b], show that U(f) = L(f) = 0.
Let f(x) = x2 for 0 < x < 1. For the partition Pn := (0, 1/n, 2/n, . . . , (n - 1)/n, 1), calculate L(f, Pn) and U(f, Pn), and show that L(f) = U(f) = 1/3. (Use the formula 12 + 22 + ∙ ∙ ∙+m2 = 1/6m(m + 1)(2m + 1).)
Let Pε be the partition whose existence is asserted in the Integrability Criterion 7.4.8. Show that if P is any refinement of Pε, then U(f; P) - L(f; P) < ε.
Write out the proofs that a function f on [a, b] is Darboux integrable if it is either(a) Continuous.(b) Monotone.
Let f be defined on I := (a, b) and assume that f satisfies the Lipschitz condition |f(x) - f(y)| < K|x - y| for all x, y in I. If Pn is the partition of I into n equal parts, show that 0 < U (f; Pn) - ∫ba f < K(b - a)2/n:
Prove if f(x) := c for x ∈ (a, b), then its Darboux integral is equal to c(b - a).
Let f and g be bounded functions on I := (a, b). If f(x) < g(x) for all x ∈ I, show that L(f) < L(g) and U(f) < U(g).
(a) Prove that if g(x) := 0 for 0 < x < 1/2 and g(x) := 1 for 1/2 < x < 1, then the Darboux integral of g on [0, 1] is equal to 1/2. (b) Does the conclusion hold if we change the value of g at the point 1/2 to 13?
Let f be continuous on I := (a, b) and assume f(x) > 0 for all x ∈ I. Prove if L(f) = 0, then f(x) = 0 for all x ∈ I.
Let f1 and f2 be bounded functions on (a, b). Show that L(f1) + L(f2) < L(f1 + f2).
Note that ∫10 (1 - x2)1/2 dx = p/4. Explain why the error estimates given by formulas (4), (7), and (10) cannot be used. Show that if h(x) = (1 - x2)1/2 for x in [0, 1], then Tn(h) < π/4 < Mn(h). Calculate M8(h) and T8(h).
Show that if fʹʹ(x) > 0 on (a, b) (that is, if f is convex on [a, b]), then for any natural numbers m, n we have Mn(f) < ∫ba f(x)dx < Tm(f): If fʹʹ(x) < 0 on [a, b], this inequality is reversed.
Show that lim((cos πx)2n) exists for all x ∈ R. What is its limit?
Show that if a > 0, then the convergence of the sequence in Exercise 6 is uniform on the interval [a, ∞), but is not uniform on the interval (0, ∞).
Show that if a > 0, then the sequence (n2x2e-nx) converges uniformly on the interval [a, ∞), but that it does not converge uniformly on the interval [0, ∞).
Show that if (fn), (gn) converge uniformly on the set A to f, g, respectively, then (fn + gn) converges uniformly on A to f + g.
Show that if fn(x) := x + 1/n and f(x) := x for x ∈ R, then (fn) converges uniformly on R to f, but the sequence (f2n) does not converge uniformly on R. (Thus the product of uniformly convergent sequences of functions may not converge uniformly.)
Let (fn), (gn) be sequences of bounded functions on A that converge uniformly on A to f, g, respectively. Show that (fngn) converges uniformly on A to fg.
Let (fn) be a sequence of functions that converges uniformly to f on A and that satisfies |fn (x)| < M for all n ∈ N and all x ∈ A. If g is continuous on the interval = (-M; M), show that the sequence (g o fn) converges uniformly to g o f on A.
Evaluate lim(xn/(1 + xn)) for x ∈ R, x > 0.
Show that lim(Arctan nx) = (π/2)sgn x for x ∈ R.
Show that lim x2e-nx) = 0 and that lim (n2x2e-nx) = 0 for x ∈ R; x > 0.
Show that the sequence (xn/(1 + xn)) does not converge uniformly on [0, 2] by showing that the limit function is not continuous on [0, 2].
Let gn(x) := e-nx/n for x > 0, n ∈ N. Examine the relation between lim(gn) and lim(gʹn).
If a > 0, show that lim ∫πa (sin nx)/(nx)dx = 0. What happens if a = 0?
Let fn(x) := nx/(1 + nx) for x ∈ (0, 1). Show that (fn) converges non uniformly to an integrable function f and that ∫10 f(x)dx = lim ∫10 fn(x)dx.
Let gn(x) := nx(1 - x)n for x ∈ (0, 1), n ∈ N. Discuss the convergence of (gn) and (∫10 gndx).

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