Intereting Posts

Is $7k-9$ ever a power of $2$?
Existence of finite indexed normal subgroup for a given finite indexed subgroup.
Distinct digits in a combination of 6 digits
Why do we essentially need complete measure space?
How to integrate $ \int_0^\infty \sin x \cdot x ^{-1/3} dx$ (using Gamma function)
Characterization of Harmonic Functions on the Punctured Disk
Find all solutions to $x^9 \equiv 25$(mod 29)
On a topological proof of the infinitude of prime numbers.
A question about a proof of the “Least Upper Bound Property” in the Tao's Real Analysis notes
Show that every nonzero integer has balanced ternary expansion?
Completion and algebraic closure commutable
Real Analysis, Folland Problem 6.1.2 $L^p$ spaces
Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
If |f| is Riemann integrable, then f is Riemann integrable???

Do there exist smooth $f,g : \mathbb{R} \to \mathbb{R}$ such that $f \neq g$, but $f(x) = g(x)\ \ \forall\ x \in (a, b)$ (assume $a < b$)?

- Difficult improper integral: $\int_0^\infty \frac{x^{23}}{(5x^2+7^2)^{17}}\,\mathrm{d}x$
- Find the limit of $(2\sin x-\sin 2x)/(x-\sin x)$ as $x\to 0$ without L'Hôpital's rule
- Derivative of matrix exponential wrt to each element of Matrix
- Why we use dummy variables in integral?
- Deriving the addition formula for the lemniscate functions from a total differential equation
- Can an ordered field be finite?
- Evaluating $\lim _{x\to 1}\left(\frac{\sqrt{x}-1}{2\sqrt{x}-2}\right)$
- Is there a formula for $\sin(xy)$
- A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$
- What are some easy to understand applications of Banach Contraction Principle?

Canonical example:

$$f(x) = \begin{cases} e^{-\frac 1x} &\text{if }x>0 \\ 0 &\text{if }x\le 0\end{cases}$$

is smooth and equal to the zero function on the interval $(-\infty, 0)$, but is not the zero function.

- Quasi-interactive proof on real numbers
- Show that $\phi(mn) = \phi(m)\phi(n)\frac{d}{\phi(d)}$
- Permutation count of AABBC
- The derivation of the Weierstrass elliptic function
- Find solution of equation $(z+1)^5=z^5$
- construct polynomial from other polynomials
- Product of principal ideals: $(a)\cdot (b) = (a b)$
- Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$
- Show that if $ab \equiv ac$ mod $n$ and $d=(a,n)$, then $b \equiv c$ mod $\frac{n}{d}$
- A game with $\delta$, $\epsilon$ and uniform continuity.
- Anti-compact space
- Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
- What is an example of real application of cubic equations?
- Proving $\cos(x)^2+\sin(x)^2=1$
- Prove that the operator norm is a norm