Theoretical Exercises on Calculus II

[Tolaso J Kos]

1. The function \( f \) is defined as \( f:[0, 1] \rightarrow \mathbb{R} \) by $$f(x)=\left\{\begin{matrix}
0\,, &x \in [0,1]\cap \left ( \mathbb{R}\setminus \mathbb{Q} \right ) \\
x_n\,, & x=q_n \in [0, 1]\cap \mathbb{Q}
\end{matrix}\right.$$ whereas \( x_n \) is a sequence such that \( \lim x_n =0 \) and \( 0\leq x_n \leq 1 \) for all \( n \) and \( q_n \) such that \( \mathbb{Q}\cap \left [ 0, 1 \right ]=\left \{ q_n, \;\; n \in \mathbb{N} \right \} \) (an enumeration of the rationals in the unit interval).

Prove that \( f \) is Riemann integrable and that \( \displaystyle \int_{0}^{1}f(x)\,dx=0 \).


2. If \( f \) is continous on \( [a, b ] \) prove that: $$\int_{a}^{b}f(x_1)\,dx_1\int_{a}^{x_1}f(x_2)\,dx_2\cdots\int_{a}^{x_{n-1}}f(x_n)\,dx_n=\frac{1}{n!}\left [ \int_{a}^{b}f(x)\,dx \right ]^n$$