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

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