Chapter 1

Riemann Integration

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Section 1AReview: Riemann Integral[14/0%]
Problem 1A.1

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is a bounded function such that

L(f,P,[a,b])=U(f,P,[a,b]) L(f, P,[a, b]) = U(f, P,[a, b])

for some partition PP of [a,b][a, b]. Prove that ff is a constant function on [a,b][a, b].

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Problem 1A.2

Suppose a≤s<t≤ba \leq s < t \leq b. Define f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} by

f(x)={1 if s<x<t0 otherwise  f(x) = \begin{cases} 1 & \text{ if } s < x < t \\ 0 & \text{ otherwise }\end{cases}

Prove that ff is Riemann integrable on [a,b][a, b] and that ∫abf=t−s\int_{a}^{b} f = t-s.

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Problem 1A.3

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is a bounded function. Prove that ff is Riemann integrable if and only if for each ε>0\varepsilon > 0, there exists a partition PP of [a,b][a, b] such that

U(f,P,[a,b])−L(f,P,[a,b])<ε. U(f, P,[a, b])-L(f, P,[a, b]) < \varepsilon .
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Problem 1A.4

Suppose f,g:[a,b]→Rf, g:[a, b] \rightarrow \mathbb {R} are Riemann integrable. Prove that f+gf+g is Riemann integrable on [a,b][a, b] and

∫ab(f+g)=∫abf+∫abg. \int _{a}^{b}(f+g) = \int _{a}^{b} f+\int _{a}^{b} g.
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Problem 1A.5

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is Riemann integrable. Prove that the function −f-f is Riemann integrable on [a,b][a, b] and

∫ab(−f)=−∫abf \int _{a}^{b}(-f) = -\int _{a}^{b} f
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Problem 1A.6

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is Riemann integrable. Suppose g:[a,b]→Rg:[a, b] \rightarrow \mathbb {R} is a function such that g(x)=f(x)g(x) = f(x) for all except finitely many x∈[a,b]x \in [a, b]. Prove that gg is Riemann integrable on [a,b][a, b] and

∫abg=∫abf \int _{a}^{b} g = \int _{a}^{b} f
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Problem 1A.7

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is a bounded function. For n∈Nn \in \mathbb {N}, let PnP_{n} denote the partition that divides [a,b][a, b] into 2n2^{n} intervals of equal size. Prove that

L(f,[a,b])=lim⁡n→∞L(f,Pn,[a,b]) and U(f,[a,b])=lim⁡n→∞U(f,Pn,[a,b]) L(f,[a, b]) = \lim _{n \rightarrow \infty } L\left(f, P_{n},[a, b]\right) \text{ and } U(f,[a, b]) = \lim _{n \rightarrow \infty } U\left(f, P_{n},[a, b]\right)
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Problem 1A.8

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is Riemann integrable. Prove that

∫abf=lim⁡n→∞b−an∑j=1nf(a+j(b−a)n). \int _{a}^{b} f = \lim _{n \rightarrow \infty } \frac{b-a}{n} \sum _{j = 1}^{n} f\left(a+\frac{j(b-a)}{n}\right).
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Problem 1A.9

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is Riemann integrable. Prove that if c,d∈Rc, d \in \mathbb {R} and a≤c<d≤ba \leq c < d \leq b, then ff is Riemann integrable on [c,d][c, d].

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Remark.
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To say that ff is Riemann integrable on [c,d][c, d] means that ff with its domain restricted to [c,d][c, d] is Riemann integrable.

Problem 1A.10

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is a bounded function and c∈(a,b)c \in (a, b). Prove that ff is Riemann integrable on [a,b][a, b] if and only if ff is Riemann integrable on [a,c][a, c] and ff is Riemann integrable on [c,b][c, b]. Furthermore, prove that if these conditions hold, then

∫abf=∫acf+∫cbf \int _{a}^{b} f = \int _{a}^{c} f+\int _{c}^{b} f
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Problem 1A.11

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is Riemann integrable. Define F:[a,b]→RF:[a, b] \rightarrow \mathbb {R} by

F(t)={0 if t=a∫atf if t∈(a,b] F(t) = \begin{cases} 0 & \text{ if } t = a \\ \int _{a}^{t} f & \text{ if } t \in (a, b]\end{cases}

Prove that FF is continuous on [a,b][a, b].

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Problem 1A.12

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is Riemann integrable. Prove that ∣f∣\left|f\right| is Riemann integrable and that

∣∫abf∣≤∫ab∣f∣ \left|\int _{a}^{b} f\right| \leq \int _{a}^{b}\left|f\right|
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Problem 1A.13

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is an increasing function, meaning that c,d∈[a,b]c, d \in [a, b] with c<dc < d implies f(c)≤f(d)f(c) \leq f(d). Prove that ff is Riemann integrable on [a,b][a, b].

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Problem 1A.14

Suppose f1,f2,…f_{1}, f_{2}, \ldots is a sequence of Riemann integrable functions on [a,b][a, b] such that f1,f2,…f_{1}, f_{2}, \ldots converges uniformly on [a,b][a, b] to a function f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R}. Prove that ff is Riemann integrable and

∫abf=lim⁡n→∞∫abfn \int _{a}^{b} f = \lim _{n \rightarrow \infty } \int _{a}^{b} f_{n}
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Section 1BRiemann Integral Is Not Good Enough[5/0%]
Problem 1B.1

Define f:[0,1]→Rf:[0,1] \rightarrow \mathbb {R} as follows:

f(a)={0 if a is irrational 1n if a is rational and n is the smallest positive integer  such that a=mn for some integer m f(a) = \begin{cases} 0 & \text{ if } a \text{ is irrational } \\ \frac{1}{n} & \text{ if } a \text{ is rational and } n \text{ is the smallest positive integer } \\ & \text{ such that } a = \frac{m}{n} \text{ for some integer } m\end{cases}

Show that ff is Riemann integrable and compute ∫01f\int_{0}^{1} f.

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Problem 1B.2

Suppose f:[a,b]→Rf:[a, b] \rightarrow \mathbb {R} is a bounded function. Prove that ff is Riemann integrable if and only if

L(−f,[a,b])=−L(f,[a,b]) L(-f,[a, b]) = -L(f,[a, b])
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Problem 1B.3

Suppose f,g:[a,b]→Rf, g:[a, b] \rightarrow \mathbb {R} are bounded functions. Prove that

L(f,[a,b])+L(g,[a,b])≤L(f+g,[a,b]) L(f,[a, b])+L(g,[a, b]) \leq L(f+g,[a, b])

and

U(f+g,[a,b])≤U(f,[a,b])+U(g,[a,b]) U(f+g,[a, b]) \leq U(f,[a, b])+U(g,[a, b])
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Problem 1B.4

Give an example of bounded functions f,g:[0,1]→Rf, g:[0,1] \rightarrow \mathbb {R} such that

L(f,[0,1])+L(g,[0,1])<L(f+g,[0,1]) L(f,[0,1])+L(g,[0,1]) < L(f+g,[0,1])

and

U(f+g,[0,1])<U(f,[0,1])+U(g,[0,1]) U(f+g,[0,1]) < U(f,[0,1])+U(g,[0,1])
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Problem 1B.5

Give an example of a sequence of continuous real-valued functions f1,f2,…f_{1}, f_{2}, \ldots on [0,1][0,1] and a continuous real-valued function ff on [0,1][0,1] such that

f(x)=lim⁡k→∞fk(x) f(x) = \lim _{k \rightarrow \infty } f_{k}(x)

for each x∈[0,1]x \in [0,1] but

∫01f≠lim⁡k→∞∫01fk \int _{0}^{1} f \neq \lim _{k \rightarrow \infty } \int _{0}^{1} f_{k}
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