What Happened 11/28

Class on Tuesday 11/28 began with student presentations of solutions to the Homework 9 challenge problems.

We then made the proposition that if $f:[a,b]\to R$ is non-decreasing (i.e. $f(x)\leq f(y)\;\forall x<y\in[a,b]$ , then $\int_a^b f$ exists.

We then produced an outline of how to prove it, which essentially consisted of 3 steps:

a) choose arbitrary $\epsilon>0$

b) choose $P_\epsilon\in P_{[a,b]}$ , with a condition that we determined to be $(x_k-x_{k-1}<\frac{\epsilon}{f(b)-f(a)}$

c) $U(f,P_\epsilon)-L(f,P_\epsilon)$

We then produced a partial proof of the following:

If $g:[0,1]\to[0,1]$ is bijective and increasing, then $\int_0^1g+\int_0^1g^{-1}=1$

We first looked at a geometric argument, by drawing an example graph, to show that the sum was in fact 1.

Then, we proved that the integrals on the left side exist.

$\int_0^1g$ followed directly from the proposition we provide the proof outline for

We then showed that $g^{-1}$ must be increasing on the interval, which then allows us to apply the proposition

Finally, we went over the statement of the Fundamental Theorem of Calculus.

Math 320: Real Analysis