## Analysis I: Lecture 20

In which we study some basic properties of the integral.

• Theorem 48 (Criterion for Riemann integrability) A bounded function $f : [a,b] \to \mathbb{R}$ is Riemann integrable if and only if given any $\epsilon > 0$ there is a dissection $\mathcal{D}$ with $S(f,\mathcal{D}) - s(f,\mathcal{D}) < \epsilon$We proved this from the definition of Riemann integration.
• Lemma 49
1. The function $J:[a,b] \to \mathbb{R}$ defined by $J(t) = 1$ for all $t$ is integrable, and $\int_a^b 1 \mathrm{d}x = b-a$.
2. If $f$, $g : [a,b] \to \mathbb{R}$ are integrable, then so is $f+g$, and $\int_a^b f(x) + g(x) \mathrm{d}x = \int_a^b f(x) \mathrm{d}x + \int_a^b g(x) \mathrm{d}x$.
3. If $f:[a,b] \to \mathbb{R}$ is integrable, then so is $-f$, and $\int_a^b (-f(x)) \mathrm{d}x = -\int_a^b f(x) \mathrm{d}x$.
4. If $f:[a,b] \to \mathbb{R}$ is integrable, then so is $\lambda f$ for any $\lambda \in \mathbb{R}$, and $\int_a^b \lambda f(x) \mathrm{d}x = \lambda \int_a^b f(x) \mathrm{d}x$.
5. If $f$, $g:[a,b] \to \mathbb{R}$ are integrable and $f(t) \geq g(t)$ for all $t \in [a,b]$, then $\int_a^b f(x) \mathrm{d}x \geq \int_a^b g(x) \mathrm{d}x$.

Theorem 48 was rather useful for this.

• Lemma 50
1. If $f:[a,b] \to \mathbb{R}$ is integrable then so is $f^2$.
2. If $f$, $g:[a,b] \to \mathbb{R}$ are integrable, then so is the pointwise product $fg$.

We’ll prove this next time.

#### Understanding today’s lecture

Hopefully you’ve tried to use the definition of Riemann integration to show that certain functions are or are not Riemann integrable.  What happens if you try to use Theorem 48?  Are there functions for which it’s easier to use the definition?  Are there functions for which it’s easier to use Theorem 48?

We’re in the midst of doing stuff with the Riemann integral, so the same comments apply here as in the last post. Inevitably Wikipedia has a page about Riemann integration.

#### Preparation for Lecture 21

Can you prove the first part of Lemma 50 (using our criterion for integrability or in any other way)?  Can you deduce the second part from the first?

Is an increasing function $f:[a,b] \to \mathbb{R}$ necessarily Riemann integrable?

Is a continuous function $f:[a,b] \to \mathbb{R}$ necessarily Riemann integrable?