IX. IVT & Countability #

Problems that combine continuity with density, the Intermediate Value Theorem, compactness, and cardinality arguments.

Problem 1 #

Consider a function $f: \mathbb{R} \to \mathbb{R}$ which satisfies $f(x) = 0$ for all $x \in \mathbb{Q}$. Show that if $f$ is continuous, then in fact $f(x)=0$ for all $x \in \mathbb{R}$!

Solutions: Hint · Solution · Long solution

Problem 2 #

This problem revisits the notion of an additive homomorphism, this time with a slightly enlarged domain. Suppose that $f: \mathbb{R} \to \mathbb{R}$ is an additive homomorphism; that is, it satisfies

$$f(x+y)=f(x)+f(y) \text{ for all } x,y \in \mathbb{R}.$$

Describe the set of all possibilities for $f$ if we assume that it is continuous.

Extra Credit: What are the possibilities for $f$ if we don’t assume that it is continuous?

Solutions: Hint · Solution · Long solution

Problem 3 #

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is continuous and further suppose that $f(x)$ is rational whenever $x$ is irrational. Prove or disprove: $f$ must be constant.

Hint: If $f$ is not constant, what must the cardinality of its image be? Use the Intermediate Value Theorem. Also, recall that the rationals are countable and any interval is not.

Solutions: Hint · Solution · Long solution

Problem 4 #

It is possible to show that the subsets $(0,1)$ and $[0,1]$ of $\mathbb{R}$ are equinumerous, that is, there is a bijection between them.

(a) Does there exist a continuous bijection $f: [0,1] \to (0,1)$? Remember that $[0,1]$ is compact.

(b) Show that a bijection $f: (0,1) \to [0,1]$ cannot be continuous. Hint: Use the Intermediate Value Theorem.

Solutions: Hint · Solution · Long solution

Problem 5 #

Suppose that $f: S_1 \to S_2$ is a continuous bijection between metric spaces $(S_1,\rho_1)$ and $(S_2, \rho_2)$. Since $f$ is a bijection, it has an inverse.

Question: Is $f^{-1}$ continuous as well?

There are certain important settings when the answer is “yes.” For instance, if $(S_1,\rho_1)=(S_2, \rho_2)=(\mathbb{R}, |\cdot|)$ this result is called the Invariance of Domain Theorem. Unfortunately, there are settings where the answer may be “no.” Find metric spaces $S_1$ and $S_2$ and a continuous bijection between them whose inverse is not continuous.

Hint: One possible example uses a discrete metric space.

Solutions: Hint · Solution · Long solution