Intereting Posts

Determinant of matrix composition
Intersection of ellipse and hyperbola at a right angle
Do diffeomorphisms act transitively on a manifold?
Distinguishing between symmetric, Hermitian and self-adjoint operators
Matrix with non-negative eigenvalues (and additional assumption)
Finite Abelian groups: $G \times H \cong G\times K$ then $H\cong K$
Solving modular arithmetic questions
Laguerre polynomials and inclusion-exclusion
Inequalities from Taylor expansions of $\log$ functions
Intuition behind conjugation in group theory
difference between a $G$ invariant measure on $G/H$ and a haar measure on $G/H$
Sufficient and necessary conditions for $f,g$ such that $\Bbb F_m/(f) \cong \Bbb F_m/(g)$
$\int_{-\infty}^\infty e^{ikx}dx$ equals what?
How does one show that $\int_{0}^{\pi/4}\sin(2x)\ln{(\ln^2{\cot{x}})}\mathrm dx=\ln{\pi\over4}-\gamma?$
Understanding the Proof of Dirac's Theorem Regarding Graph Connectivity

Let $f:\Bbb R^2\to \Bbb R$ be a continuously differentiable function. Show that $f$ is not $1-1$.

I know I will need to use the Inverse Function Theorem and consider some open set A with $g:A\to \Bbb R^2$ defined by $g(x,y)=\big(f(x,y),y\big)$.

- Pointwise convergence does not imply $f_n(x_n)$ converges to $f(x)$
- How to show that $f$ is a straight line if $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$?
- Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
- Upper semicontinuous functions
- About the continuity of the function $f(x) = \sum\limits_k2^{-k}\mathbf 1_{q_k \leq x}$
- Existence of Zero Divisors in $C(X,\mathbb{R})$

- If $\,\lim_{x\to 0} \Big(f\big({a\over x}+b\big) - {a\over x}\,f'\big({a\over x}+b\big)\Big)=c,\,$ find $\,\lim_{x\to\infty} f(x)$
- How to proof the the derivative of $x^2$ is $2x$?
- Understanding second derivatives
- Tracing a curve along itself - can the result have holes?
- a function $f$ is differentiable in $\vec{0}$ if $f \circ \gamma $ is differentiable in 0
- Proof for $\sin(x) > x - \frac{x^3}{3!}$
- Existence of continuous angle function $\theta:S^1\to\mathbb{R}$
- $x(a^{1/x}-1)$ is decreasing
- What can you say about a continuous function that is zero at all integer values?
- limits of integration and derivative

Being continuous, $f$ assumes its min and max on the compact $S^1$. By the IVT, any value between min and max ias assumed once in each of the two arcs bounded by a minimal and a maximal point.

Correct me if I’m wrong, but I thought this was a more general result:

No function defined on a compact subset of $\mathbb{R}^2$ can be 1-1, basically by the pigeonhole principle.

The nonexistence of a utility function representing lexicographic preferences basically boils down to this principle.

- Does path-connected imply simple path-connected?
- How are $G$-modules and linear group actions different
- Why does “the probability of a random natural number being prime” make no sense?
- About 0.999… = 1
- Mean value of arithmetic function
- Graph theory and tree company
- Does the sum $\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}$ converge for $\alpha > \frac{1}{2}$ and $x \in $?
- $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample
- Do trivial homology groups imply contractibility of a compact polyhedron?
- A rigorous book on a First Course in linear algebra
- $L^p$ submartingale convergence theorem
- Integrate: $ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $ without using complex analysis methods
- Algorithm to determine matrix equivalence
- Division by $0$
- Applying Extended Euclidean Algorithm for Galois Field to Find Multiplicative Inverse