Intereting Posts

Differentiate the Function: $y=x^x$
How is this subgroup a normal subgroup?
expected number of cards drawn exactly once (with replacement)
A function can provide the complete set of Euler primes via a Mill's-like constant. Is it useful or just a curiosity?
Alternative form to express the second derivative of $\zeta (2) $
Proof that $a\equiv 1\,(\textrm{mod }8)$ implies $a$ is a square modulo $2^n$ for all $n$
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$
Calculating the area of an irregular polygon
Quotient space of closed unit ball and the unit 2-sphere $S^2$
Solving very large matrices in “pieces”
Relations between the maximum matching, minimum vertex cover, maximum independent set, and maximum vertex biclique for a bipartite graph
Brownian motion martingale
Given the first $n$ terms of a sequence, can we find a function which would generate any infinite sequence we like that has the same first $n$ terms?
Why morphism between curves is finite?
What is the difference between “family” and “set”?

For $a<b, a,b\in\Bbb R$

$$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta] $$

Clearly the RHS is an (uncountable) infinite sum of closed intervals. I have no idea how to show it is open at two ends.

- Equivalence of measures and $L^1$ functions
- If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
- Is $f(x)=\sup_{y\in K}g(x, y)$ a continuous function?
- Question about Geometric-Harmonic Mean.
- Forcing series convergence
- Connected Implies Path (Polygonally) Connected

(My hope is that if this is true, then it is trivial that:

$$f(x)\in\mathscr C^r(a,b)\Longleftrightarrow f(x)\in\mathscr C^r[a+\delta,b-\delta]\quad \forall \delta\in(0,\frac{b-a}2) $$

which will be very useful for me.

A proof that is neat (without too much set theoretical jargon) and based on the original definition will be sincerely appreciated. Thanks in advance!

EDIT. Thanks for Surb’s answer. Now I understand why LHS and RHS are equal. But I’m still confused about another question:

Since they two are equal, doesn’t it imply that if a statement A is true on LHS then it is also true on RHS? However, there are many counter-examples. Like uniform continuity for a function, or uniform convergence for a functional sequence on RHS doesn’t imply that on LHS, but… How come?

- Example that a measurable function $f$ on $[1,\infty )$ can be integrable when $\sum _{n=1}^{\infty }\int_{n}^{n+1}f$ diverges.
- Prove that if a set $E$ is closed iff it's complement $E^{c}$ is open
- If a Riemann integrable function is zero on a dense set, then its integral is zero
- What is a Real Number?
- If $\lim_{h\to 0} \frac{f(x_0 + h) - f(x_0 - h)}{2h} = f'(x_0)$ exists, is f differentiable at $x_0$?
- Twice differentiable function, show there is a fixed point
- Evaluate $\lim_{x \to \infty} \frac{(\frac x n)^x e^{-x}}{(x-2)!}$
- Composition of Riemann integrable functions
- Riemann-Stieltjes integral, integration by parts (Rudin)
- How to find a measurable but not integrable function or a positive integrable function?

if $x\in (a,b)$ there is a $\delta>0$ such that $x\in[a+\delta,b-\delta]$ and thus $(a,b)\subset \bigcup_{\delta}I_\delta$. The other inclusion is obvious.

- Closed form for improper definite integral involving trig functions and exponentials?
- Mathematical difference between white and black notes in a piano
- Integrate: $\int \frac{dx}{x \sqrt{(x+a) ^2- b^2}}$
- How to solve $\int_0^\pi \frac{x\sin x}{1+ \sin^2 x}dx$?
- How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?
- Show $ \int _0^t \frac{\left|B_u \right|}{u}du < \infty \ a.e.$
- Proving whether ideals are prime in $\mathbb{Z}$
- Simplifying Catalan number recurrence relation
- Find the eigenvalues and eigenvectors with zeroes on the diagonal and ones everywhere else.
- Notation regarding different derivatives
- Dense and locally compact subset of a Hausdorff space is open
- linear transformation $T$ such that $TS = ST$
- Integral of the derivative of a function of bounded variation
- Linear functional on a Banach space is discontinuous then its nullspace is dense.
- Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)