Intereting Posts

Fourier transform of $\operatorname{erfc}^2\left|x\right|$
A limit question (JEE $2014$)
Evaluate integral: $ \int_{-1}^{1} \frac{\log|z-x|}{\pi\sqrt{1-x^2}}dx$
If $\mathrm{E} |X|^2$ exists, then $\mathrm{E} X$ also exists
How prove this integral equation $\int_{a}^{b}\frac{1}{\sqrt{|f(x)|}}dx=\int_{c}^{d}\frac{1}{\sqrt{|f(x)|}}dx$
Comparing Eigenvalues of Positive Semidefinite Matrices
2 color theorem
How to find the minimum value of $|5^{4m+3}-n^2 |$
Mid-point convexity does not imply convexity
What method would you go through to find an $n$ such that $\phi(n)=100$?
Splitting of the tangent bundle of a vector bundle and connections
Sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$
Full algorithm for rational functions integration
hint to find the second derivative
How to prove this inequality(7)?

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?

- Is there a topology on the full transformation semigroup?
- The Stone-Čech compactification of a space by the maximal ideals of the ring of bounded continuous functions from the space to $\mathbb{R}$
- To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous
- compactness / sequentially compact
- Motivation behind topology
- manifold as simplicial complex

- Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$
- Separable implies second countable
- Is an infinite line the same thing as an infinite circle?
- Projection map being a closed map
- Show that $(X\times Y)\setminus (A\times B)$ is connected
- Infinity-to-one function
- Discrete non archimedean valued field with infinite residue field
- Doubt about proof of factorization $f=pi$, where $i$ is acyclic cofibration and $p$ is fibration
- If $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded.
- Euclidean spaces

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 $V = B_{\delta}(x)$.

- Dissecting a proof of the $\Delta$-system lemma
- Show that $p \Rightarrow (\neg(q \land \neg p))$ is a tautology
- Solve $(n+1)f(n+1)=(n+2)f(n)+1$
- Using definition of derivative to differentiate $f(x)= \sqrt{x}+1$
- Basic problem on topology $( James Dugundji)$
- Finding the fundamental group of the complement of a certain graph
- how to find mid point of an arc?
- If $f$ is Riemann-Stieltjes Integrable, then does there exist a partition of which each lengths of subinterval are the same?
- Random Variables and Moment Generating functions
- How many countable graphs are there?
- Is the matrix diagonalizable for all values of t?
- Sieving integers
- Relationship between Dixonian elliptic functions and Borwein cubic theta functions
- A limit involves series and factorials
- Special arrows for notation of morphisms