Intereting Posts

Give an example of a continuous function $f:R\rightarrow R$ which attains each of its values exactly three times.
Product of two Lebesgue integrable functions not Lebesgue integrable
Bounds for lcm$(1,\dots,n)$
Are there any calculus/complex numbers/etc proofs of the pythagorean theorem?
Finding an equation for a circle given its center and a point through which it passes
Why is the localization of a commutative Noetherian ring still Noetherian?
Proving Stirling formula for complex variable
What is the norm of the operator $L((x_n)) \equiv \sum_{n=1}^\infty \frac{x_n}{\sqrt{n(n+1)}}$ on $\ell_2$?
Does the Riemann tensor encode all information about the second derivatives of the metric?
Results that were widely believed to be false but were later shown to be true
how to find $\int_{0}^{1}h_n(x)dx?$
Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^32^n}$
limit of $\sqrt{\frac{1\cdot 3\cdots (2n-1)}{2\cdot 4\cdots (2n)}}$ using geometric mean
Very tight prime bounds
Prove sum of primitive roots congruent to $\mu(p-1) \pmod{p}$

Show that $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous in the $\delta-\epsilon$ definition of continuity if and only if for all $x \in \mathbb{R}$ and all open set $U$ where $f(x) \in U$, there exists an open set $V$ where $x \in V$ and $f(V) \subset U$

Forward direction: Let open set $V$ such that $V \subset B_{\delta}(x)$. Then we have $f(V) \subset f(B_{\delta}(x)) \subset B_{\epsilon}(f(x)) \subset U$. The last inclusion follows by the openness of $U$ and also $f(x) \in U$

Backward direction: No idea. Can anyone guide me?

- Homeomorphism preserving distance
- Understanding of nowhere dense sets
- What is $(S^1\times S^1)/C_{n}$ topologically?
- Derived sets - prove $(A \cup B)' = A' \cup B'$
- Is the result true when the valuation is trivial and $\dim(X)=n$?
- Prob. 4, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: For $T_1$-spaces countable compactness is equivalent to limit-point-compactness.

- What application is there for a non-Hausdorff topological space?
- sequentially continuous on a non first-countable
- Finding counterexamples: bijective continuous functions that are not homeomorphisms
- Is projection of a closed set $F\subseteq X\times Y$ always closed?
- Is the result true when the valuation is trivial and $\dim(X)=n$?
- How to prove that $ \text{int}(\text{cl}(A)) = \text{cl}(\text{int}(A)) $?
- Continuity and image of convergent sequences
- Quotient Space of Hausdorff space
- Continuous bijection from $\mathbb{R}^n$ to $\mathbb{R}^m$
- Union of connected subsets is connected if intersection is nonempty

Let $x$ be such that $f(x) \in U$. Then by assumption that $U$ is open, there is some $\epsilon > 0$ such that $B_{\epsilon} f(x) \subset U$. Find $\delta >0$ such that $y \in B_{\delta}(x) \implies f(y) \in B_{\epsilon}f(x)$ (why can we find such a $\delta$?). Now, take $V = B_{\delta}(x)$.

- Bell numbers and moments of the Poisson distribution
- “Graded free” is stronger than “graded and free”?
- Inverting the Cantor pairing function
- If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax-by=1$.
- How to prove that the Kronecker delta is the unique isotropic tensor of order 2?
- A golden ratio series from a comic book
- On the sum of digits of $n^k$
- What's the difference between a monoid and a group?
- Prove that integral of continuous function is continuously differentiable
- The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$
- Increasing real valued function whose image set is connected
- How to prove this maximum is $\frac{\sqrt{3}}{5}$
- Sum up to number $N$ using $1,2$ and $3$
- Recommendations for Intermediate Level Logics/Set Theory Books
- Graph of continuous function from compact space is compact.