# Muddiest Point 11/16

In this edition of the muddiest point I will go over non-Riemann integrable functions. Let’s review a classic example that we mentioned in class, the Dirichlet function. Define the Dirichlet Function as
$$f(x)=\begin{cases} 1\ \text{if}\ x\in\mathbb{Q}\\ 0\ \text{if}\ x\notin\mathbb{Q}\end{cases}$$
Recall that by definition, A bounded function $f$ defined on the interval $[a,b]$ is Riemann-integrable if $U(f)=L(f)$. For $\mathbb{P}$, the set of all possible partitions of $[a,b]$, we call $U(f)=\inf\{U(f,P):P\in\mathbb{P}\}$ the upper integral of $f$. Similarly, $L(f)=\sup\{L(f,P):P\in\mathbb{P}\}$ is the lower integral of $f$. (Notice that these use the lower and upper sums that we discussed in class. We define the lower sum $L(f,P)=\sum_{k=1}^n m_k(x_k-x_{k-1})$ and the upper sum $U(f,P)=\sum_{k=1}^n M_k(x_k-x_{k-1})$).

Well, looking at the Dirichlet function, we can see that $U(f)=1$, but $L(f)=0$. It follows that $1\neq 0$, so the function is not Reimann integrable. This way of showing that the upper and lower integrals of a functions are not equal will most likely be one of our main tools to show that a function is not Riemann integrable.

#### One Response

1. Jeremy LeCrone says:

Good post. Thank you