Intereting Posts

How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x – \frac{1}{x}\right)dx?$
Definition of an Ordered Pair
Slightly changing the formal definition of continuity of $f: \mathbb{R} \to \mathbb{R}$?
Proving the limit of a recursive sequence
Proof that $6^n$ always has a last digit of $6$
For a topological group $G$ and a subgroup $H$, is it true that $ = \overline{}$? What about algebraic groups?
What is an odd prime?
What are some alternative definitions of vector addition and scalar multiplication?
A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$
Reducibility of a polynomial in $\mathbb R$
Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
Reference For Integrals
When do I use “arbitrary” and/or “fixed” in a proof?
Distance between triangle's centroid and incenter, given coordinates of vertices
How to represent XOR of two decimal Numbers with Arithmetic Operators

Is there an continous function $f: \mathbb R^2 \to \mathbb R$ such that $f^{-1}(a)$ is finite for every $a \in \mathbb R$?

It’s not possible for analytic or smooth but I’m curious about continous mapping.

- Open subspaces of locally compact Hausdorff spaces are locally compact
- Which spheres are fiber bundles?
- When is $C_0(X)$ separable?
- Separable regular Hausdorff spaces have a basis of cardinality $\le 2^{\aleph_0}$? Why is $^A$ separable iff $|A| \le 2^{\aleph_0}$?
- Show that the countable product of metric spaces is metrizable
- Bourbaki exercise on connected sets

- Maps of discs into surfaces
- Quotient Space $\mathbb{R} / \mathbb{Q}$
- Countable Connected Hausdorff Space
- Are contractible open sets in $\mathbb{R}^n$ homeomorphic to $\mathbb R^n$?
- Is there a function whose inverse is exactly the reciprocal of the function, that is $f^{-1} = \frac{1}{f}$?
- Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.
- Topologies in a Riemannian Manifold
- Prove that a continuous image of a closed subset of a compact space is a closed subset
- What are the convergent sequences in the cofinite topology
- closure of finite unions

Take any continuous map $f: \Bbb R^2 \to \Bbb R$. Suppose for contradiction that $f$ has finite fibers…

Choose two points $a,b \in f(\Bbb R^2)$ with $a < b$. We can do this because $f$ is non-constant, otherwise it cannot have finite fibers.

Pick a point $c \in (a,b)$. Then the set $A:=\Bbb R^2 \backslash \;f^{-1}(\{c\})$ is connected, since its complement is finite.

Thus $f(A)$ is a connected subset of $\Bbb R$. Since $a,b \in f(A)$ and since $c \in (a,b)$, it follows that $c \in f(A)$. But this contradicts the definition of $A$.

- What's the intuition behind the identities $\cos(z)= \cosh(iz)$ and $\sin(z)=-i\sinh(iz)$?
- Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$?
- How to solve $100x +19 =0 \pmod{23}$
- Classification of a kind of compact sets
- Conditions for vectors to span a vector space
- Calculus Integral from Partial Fractions
- What are elements in $SU(1, 1)$?
- Any compact embedded $2$-dimensional hypersurface in $\mathbb R^3$ has a point of positive Gaussian curvature
- Example of Artinian module that is not Noetherian
- $\prod (1+a_n)$ converges iff $ \sum a_n$ converges
- Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$
- Is the topology of the p-adic valuation to the unramfied extension discrete?
- Solution to differential equation $\left( 1-\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)=0$
- Can you raise a number to an irrational exponent?
- How to prove $\vdash\neg P\to (P\to Q)$?