# Math Club

#### – This Week –

On the converse of the Intermediate Value Theorem
Suppose that $f: {\Bbb R} \to {\Bbb R}$ is a function. The Intermediate Value Theorem (IVT) asserts that if $f$ is continuous then for any real numbers $a,b,y_0$ with $a < b$ and $y_0$ between $f(a)$ and $f(b)$ there exists $x_0 \in [a,b]$ such that $f(x_0)=y_0$.

We consider whether the converse of this theorem is true:

Question: Suppose that for any real numbers $a,b$, and $y_0$ with $a < b$ and $y_0$ between $f(a)$ and $f(b)$ there exists $x_0 \in [a,b]$ such that $f(x_0)=y_0$. Does this imply that $f$ is continuous?

Even if the answer to this question is negative, perhaps we can conclude that the function $f$ is close in some sense to being continuous. We will consider this question and see what we can conclude about the continuity of $f$.

The math club meets once a week in the Winter term, on Thursday evenings for about an hour. It is intended for anyone interested in mathematics, whether they are a math major or not.

At each meeting a speaker will discuss an interesting idea in mathematics. We try to pick topics that can be understood without much background. An ideal topic is one which starts with something simple, and by exploring its twists and turns, leads to something unexpectedly deeper.

The goal is to show mathematics the way that mathematicians see it: a living subject full of deep and intertwining ideas, growing naturally out of our desire to understand the fundamental concepts of space and number.

 Date Topic Speaker Jan. 26 Permutations given by Polynomials Mike Roth Feb. 2 End-to-end auditable voting systems Claude Tardif 9 When does a prime divide a binomial coefficient? Ram Murty 16 Infinitely iterated exponentials Francesco Cellarosi Mar. 2 The one-sentence proof Ivan Dimitrov 9 Some peculiarities of the three body problem. Dan Offin 16 The 25 Billion Dollar Eigenvalue Andrew McEachern 23 The Lemniscate and Generalized Trigonometric Functions Jamie Mingo 30 On the converse of the Intermediate Value Theorem David Wehlau Apr. 6 The mathematics of juggling Thomas Barthelmé

The meetings are Thursdays in Jeff 118 from 17:30–18:30.

To suggest a topic for the Math Club, or to find out further information, please email Ivan Dimitrov, Mike Roth, or Greg Smith .

 Web Site of the Week Check out Quanta Magazine; an online science magazine specializing in Mathematics, Physics, and the Life Sciences.