Intereting Posts

The index of the Core of a group
On the asymptotics of a continued fraction
Dimensions of vector subspaces
The interpretation of $0 \cdot \infty$
Removable singularities and an entire function
Convergence of series involving the prime numbers
Geometric intuition for directional derivatives
Perspective problem – trapezium turned square
If $\int_0^\infty fdx$ exists, does $\lim_{x\to\infty}f(x)=0$?
Why is $A_5$ a simple group?
Arithmetic mean. Why does it work?
Generalized Eigenvalue Problem with one matrix having low rank
How can I evaluate this limit with or without applying derivatives?
If $f'$ is increasing and $f(0)=0$, then $f(x)/x$ is increasing
Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?

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.

- Every extremally disconnected space is a perfectly $\kappa$-normal space.
- Pullback of a covering map
- Infinite product probability spaces
- Prove that if a set $E$ is closed iff it's complement $E^{c}$ is open
- If a measure only assumes values 0 or 1, is it a Dirac's delta?
- What is your definition for neighborhood in topology?

- Which of the following sets are compact:
- Understanding two similar definitions: Fréchet-Urysohn space and sequential space
- General facts about locally Hausdorff spaces?
- A question about dimension and connectedness in order topologies
- Continuous extension of a real function
- Are locally compact Hausdorff spaces with the homeomorphic one-point compactification necessarily homeomorphic themselves?
- Why is the infinite sphere contractible?
- If $A$ and $B$ are sets of real numbers, then $(A \cup B)^{\circ} \supseteq A^ {\circ}\cup B^{\circ}$
- Open maps which are not continuous
- The definition of metric space,topological space

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 $A$.

- Proving that restrictions of partial orders are partial orders
- Is my proof correct? (minimal distance between compact sets)
- Justifying the Normal Approx to the Binomial Distribution through MGFs
- Is this map uniformly continuous? continuous?
- On inequalities for norms of matrices
- Four Variable Data Processing Inequality
- How to prove the identity $3\sin^4x-2\sin^6x=1-3\cos^4x+2\cos^6x$?
- irreducible implies the commutant consists of multiples of identity?
- number of permutations in which no two consecutive numbers are adjacent
- Show that $2^n$ is not a sum of consecutive positive integers
- Pedagogy: How to cure students of the “law of universal linearity”?
- Oblique asymptotes of $f(x)=\frac{x}{\arctan x}$
- Description of $\mathrm{Ext}^1(R/I,R/J)$
- Give an example of a non-separable subspace of a separable space
- Notation on the tangent space.