Intereting Posts

Evaluate the line segment intergal
Derivative of determinant, which is correct?
Is it true that a space-filling curve cannot be injective everywhere?
Solving $|x-2| + |x-5|=3$
What are the possible eigenvalues of a linear transformation $T$ satifying $T = T^2$
A strange “pattern” in the continued fraction convergents of pi?
Problem solving Logical Equivalence Question
Show that $\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}$
Most efficient way to integrate $\int_0^\pi \sqrt{4\sin^2 x – 4\sin x + 1}\,dx$?
Proving that $\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$
Evaluating a limit involving binomial coefficients.
What do higher cohomologies mean concretely (in various cohomology theories)?
Learning how to prove that a function can't proved total?
Is there a universal property for the ultraproduct?
How can I prove that every group of $N = 255$ elements is commutative?

The $\Bbb R \to \Bbb R$ function

$$

f(x) = \begin{cases}0 & x = 0 \\

x \sin \frac{1}{x} & x \ne 0\end{cases}

$$

crosses the horizontal line $y = 0$ infinitely often. (It happens to cross many other horizontal lines infinitely often, but its restriction to any closed interval does not cross any others infinitely often, which is what interests me.)

If we define $$g(x) = f(x) + x f(x-\pi),$$ then $g$ crosses both $y = 0$ and $y = \pi$ infinitely often. Clearly, I can construct, by analogous methods, functions on closed intervals that cross any finite number of lines infinitely often. (And $f$ itself does so for every horizontal line, once we remove the restriction of the closed-interval domain.)

This motivates me to ask:

- What does recursive cosine sequence converge to?
- Show that the derivatives of a $C^1$ function vanish a.e. on the inverse image of a null set
- Intuition for uniform continuity of a function on $\mathbb{R}$
- Inverse of a composite function from $\mathbb R$ to $\mathbb{R}^p$ to $\mathbb R$ again with a non-zero continuous gradient in a point
- How to show these two definitions of the Riemann integral are equivalent?
- Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.8, Problem 3: What is the norm of this functional?

Suppose that $f : [0, 1] \to \Bbb R$ is continuous. Is it possible for the preimages of infinitely many points in $\Bbb R$ to each be infinite? Is it possible for the preimages of uncountably many points in $\Bbb R$ to be infinite?

I have a feeling that this should actually be an easy analysis question, but … it’s been too long since I studied analysis.

The motivation:

In Closed loop on the sphere is homotopic to a product of homeomorphisms onto great arcs of the sphere, the poster mistakenly believed that a space-filling curve on the sphere must “cross from $U$ to $V$ infinitely often”, where $U$ and $V$ are two open sets that form a cover of the sphere, and while I was able to address the misunderstanding, it led me to the question above.

- Proving separability of the countable product of separable spaces using density.
- When does $\sum\frac{1}{(n\ln n)^a}$ converge?
- Can a continuous real function take each value exactly 3 times?
- Given $f: \Bbb R\rightarrow\Bbb R$ and a point $a\in\Bbb R$. Prove $lim_{x\rightarrow a} f(x)=\lim_{h\rightarrow 0} f(a+h)$ if 1 of the limits exists.
- $\lim_{p\to \infty}\Vert f\Vert_{p}=\Vert f\Vert_{\infty}$?
- Prove that $ \lim\limits_{n \to \infty } \sum\limits_{k=1}^n f \left( \frac{k}{n^2} \right) = \frac 12 f'_d(0). $
- Differential Equations with Deviating Argument
- If $f(xy)=f(x)f(y)$, then $f(x)=x^b$.
- What does “sets of arbitrarily large measure” mean — question about $L_p$ embeddings
- Is there a “one-line” proof of $x<y\Rightarrow x^n<y^n$ (for $n$ an odd natural number)?

Suggestion: It seems to me that the Weierstrass function has the property that for almost every point in the codomain, the preimage is infinite. As the Wikipedia page points out, “between any two points no matter how close, the function will not be monotone.” This may be sufficient to prove the claim, or it may require a slightly more detailed analysis.

- Approximating continuous functions with polynomials
- What do modern-day analysts actually do?
- What's a good book on advanced linear algebra?
- If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$?
- Trying to understand the use of the “word” pullback/pushforward.
- Show that the derivatives of a $C^1$ function vanish a.e. on the inverse image of a null set
- Showing that $\sec z = \frac1{\cos z} = 1+ \sum\limits_{k=1}^{\infty} \frac{E_{2k}}{(2k)!}z^{2k}$
- Is there any formula for number of divisors of $a \times b$?
- Show $\nabla^2g=-f$ almost everywhere
- Evaluation of a specific determinant.
- Integrate via substitution and derivation rule
- Proving $\int_{0}^{\infty}\sin\left({1\over 4x^2}\right)\ln x\cdot{\mathrm dx\over x^2}=-\sqrt{\pi\over 2}\cdot\left({\pi-2\gamma \over 4}\right)$
- Prove that a counterexample exists without knowing one
- How many subspace topologies of $\mathbb{R}$?
- Is it possible for a function to be in $L^p$ for only one $p$?