Proving a function is not Riemann integrable

real-analysis

Hint: Every subinterval of $[0,1]$ contains both rational numbers and irrational numbers.

Look at the definition of a function being Riemann integrable on the interval $[a,b]$ (in this case $[0,1]$). This definition must fail for this function.

Why?

Well, since every subinterval contains irrational numbers, the infimum of the function values over any subinterval is $0$. Similarly, since every sub-interval contains rational numbers, the supremum of the function values over any subinterval is $1$.

What does this tell you about the upper and lower sums over any partition?

Related

Supremum and Infimum
Rationalizing a Denominator with Cube Roots
Unique expression of a polynomial under quotient mapping?
Condition on a,b and c satisfying an equation(TIFR GS 2017)
Continuum Hypothesis $\iff ?$?
Not so easy optimization of variables?
probability of getting 50 heads from tossing a coin 100 times
Help with a specific limit $\left( \dfrac{n-1}{n} \right)^n$ as $n \rightarrow \infty$
How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$?
Computing Coefficients for Generalized Combinatorial Sets
Expression for power of a natural number in terms of binomial coefficients
Prove $\sum\limits_{i=1}^{n}\frac{a_{i}^{2}}{b_{i}} \geq \frac{(\sum\limits_{i=1}^{n}a_i)^2}{\sum\limits_{i=1}^{n}b_i}$