Intereting Posts

Showing $\gcd(2^m-1,2^n+1)=1$
Nilpotent matrix and basis for $F^n$
prove $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ using index notation
Decomposition of a nonsquare affine matrix
Why is minimizing least squares equivalent to finding the projection matrix $\hat{x}=A^Tb(A^TA)^{-1}$?
Why SVD on $X$ is preferred to eigendecomposition of $XX^\top$ in PCA
Is there a proper subfield $K\subset \mathbb R$ such that $$ is finite?
Why is $\pi$ so close to $3$?
Is there a section of mathematics that studies near-integer equations.
What is the accepted syntax for a negative number with an exponent?
Any functionally complete sets with XOR?
The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$
If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$
Compute discrete logarithm
Maximal number of monomials of multivariate polynomial

Intuitively, a function $\mathbb{R}\rightarrow\mathbb{R}$ is continuous if you can draw its graph without taking the pen off the page. This suggests the following theorem:

A map $f:X \rightarrow Y$ is continuous if and only if $f$ is connected in the product topology $X \times Y$.

Is this true? And if not, can anyone think of an additional premise or two that would make it true?

- Connected metric spaces with disjoint open balls
- Show $\Bbb R ^n$ with a finite set removed is still connected.
- connected manifolds are path connected
- Does there exist a connected metric space, with more than one point and without any isolated point, in which at least one open ball is countable?
- Tracing a curve along itself - can the result have holes?
- There's no continuous injection from the unit circle to $\mathbb R$

- Definition of continuity
- Let $f:(\mathbb{R}\setminus\mathbb{Q})\cap \to \mathbb{Q}\cap $. Prove there exists a continuous$f$.
- If $f$ is continuous at $a$, is it continuous in some open interval around $a$?
- Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected
- Countablity of the set of the points where the characteristic function of the Cantor set is not continous
- Continuity of polynomials of two variables
- Discontinuous functions that are continuous on every line in $\bf R^2$
- Are there path-connected but not polygonal-connected sets?
- $f(0)=0$ and $\lvert\,f^\prime (x)\rvert\leq K\lvert\,f(x)\rvert,$ imply that $f\equiv 0$.
- For a continuous function $f$ and a convergent sequence $x_n$, lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$

It isn’t true in general. An obvious variant of the Topologist’s sine curve provides an example of a function $f:\Bbb R\rightarrow \Bbb R$ whose graph is connected but fails to be continuous (at $x=0$).

However, this article shows that “it is correct to conclude that continuous real functions over $\Bbb R$ are those functions over $\Bbb R$ whose graphs, in the plane $\Bbb R^2$, are both closed and connected”.

- Rig module (?) of measures with scalar multiplication given by Lebesgue integration
- Prove that there exist linear functionals $L_1, L_2$ on $X$
- Calculating $1+\frac13+\frac{1\cdot3}{3\cdot6}+\frac{1\cdot3\cdot5}{3\cdot6\cdot9}+\frac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $
- What is the difference between stationary point and critical point in Calculus?
- If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$
- Evaluate derivative of Lagrange polynomials at construction points
- $q$-norm $\leq$ $p$-norm
- Lüroth's Theorem
- Can't understand a proof: Let $a,b,c$ be integers. If $a$ and $b$ divide $c$, then $lcm(a,b)$ also divides $c$
- Show that there exist only $n$ solutions
- The centralizer of an element x in free group is cyclic
- Fully independent events and their complements
- Computing square roots implies factoring $n = pq$
- What number appears most often in an $n \times n$ multiplication table?
- Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$