Analysis I: Lecture 21

In which we study further properties of the integral, and find a large class of functions that are integrable.

  • 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 proved (i) using our criterion for integrability, Theorem 48, and then we deduced (ii) from (i) (despite the fact that (i) is a special case of (ii)!).

  • Lemma 51
    1. If f : [a,b] \to \mathbb{R} is integrable, then so is f_+ : [a,b] \to \mathbb{R} defined by f_+(t) = \max \{f(t),0\}.
    2. If f : [a,b] \to \mathbb{R} is integrable, then so is |f|, and \int_a^b |f| \geq |\int_a^b f|.

    For (i), we used our criterion for integrability yet again, and then we deduced (ii) from it.  This was slightly reminiscent of our proof that absolute convergence of a series implies convergence: splitting into non-negative and negative parts and then recombining to obtain the original or the absolute value of the original.

  • Lemma 52 Let f : [a,b] \to \mathbb{R} be a bounded function, and take c \in [a,b].  Then f is integrable on [a,b] if and only if f|_{[a,c]} is integrable on [a,c] and f|_{[c,b]} is integrable on [c,b].  Furthermore, if f is integrable on [a,b] then \int_a^b f = \int_a^c f|_{[a,c]} + \int_c^b f|_{[c,b]}.  The proof is an exercise.
  • Theorem 53  If f : [a,b] \to \mathbb{R} is increasing, then f is Riemann integrable.  To prove this, we used the criterion for integrability and picked a dissection of [a,b] into very narrow intervals of the same width.
  • Corollary 54
    1. If f : [a,b] \to \mathbb{R} can be written as f = f_1 - f_2 where f_1, f_2 : [a,b] \to \mathbb{R} are increasing, then f is Riemann integrable.
    2. If f : [a,b] \to \mathbb{R} is piecewise monotone, then f is Riemann integrable.

    This was straightforward using Theorem 53 and our earlier work on the basic properties of the integral.

  • I handed out the fourth examples sheet.

Understanding today’s lecture

Can you prove Lemma 52?

What other questions might you ask yourself?

Further reading

There’s an interesting Tricki article on a useful approach when trying to show that an explicit example of a function is Riemann integrable.  In fact, there’s a Tricki page at a higher level with lots of interesting thoughts about integration (and an interesting problem right at the bottom of the page).

If you’re interested, then you might like to look up uniform continuity (in a book or online), but it’s not part of this course so you can wait until Analysis II if you prefer.

Preparation for Lecture 22

Is a continuous function f : [a,b] \to \mathbb{R} necessarily integrable?  Can you justify your answer?

What is the connection between differentiation and integration?  How can we make this precise?

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3 Responses to “Analysis I: Lecture 21”

  1. guest Says:

    Can you give me a hint for the problem on the Tricki page?

  2. theoremoftheweek Says:

    I think that this is a good “Go and think about it” question, so I don’t want to spoil people’s fun by giving a hint…

  3. guest Says:

    People who want to have fun with the problem have either solved it already or are not going to read the comments… 😉

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