Math Problems Vault

Problem 1B.1

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

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 $f$ is Riemann integrable and compute $\int_{0}^{1} f$ .

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Problem 1B.2

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

L(-f,[a, b]) = -L(f,[a, b])

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Problem 1B.3

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

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

and

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 $f, g:[0,1] \rightarrow \mathbb {R}$ such that

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])

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Problem 1B.5

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

f(x) = \lim _{k \rightarrow \infty } f_{k}(x)

for each $x \in [0,1]$ but

\int _{0}^{1} f \neq \lim _{k \rightarrow \infty } \int _{0}^{1} f_{k}

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