3
votes
1answer
39 views

Function in $H^1$, but not continuous

By the Sobolev embedding theorem, if $\Omega$ is bounded, $H^s(\Omega)\subset C(\Omega)$ for $s>1$, in $\mathbb R^2$. Where I can find a counterexample (if one exists) for the case $s=1$? I mean a ...
1
vote
1answer
65 views

Example of two finitely isomorphic structures (or $\omega$-isomorphic) that are not partially isomorphic?

One can define several maps between given structures in order to portray similarities or differences. The concept of isomorphism displays a deep structural overlapping, while for instance an ...
0
votes
1answer
21 views

Convex function that has a finite limit at infinity

Can someone give me an example for a convex function that has a finit limit at infinity ?
2
votes
1answer
56 views

A question on Fourier Transform

Is there a function which is not absolutely integrable but which has a continuous fourier transform? I know that if a function is absolutely integrable then the fourier transform is continuous but I ...
2
votes
1answer
91 views

Does there exist a nowhere differentiable, everywhere continous, monotone somewhere function?

Is there a nowhere differentiable but continuous everywhere function which is monotone in some small interval however small it is? Until now I have seen only the Weierstrass function and it seems to ...
0
votes
1answer
74 views

$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?

I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might ...
7
votes
9answers
341 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...
0
votes
1answer
28 views

Simple question about montonically increasing function

Suppose we have a continuous $f: \mathbb{R} \to \mathbb{R}$, we know the definition of a monotonically increasing function is for $x,y \in \mathbb{R}$, if $x < y$ then $f(x) < f(y)$. I know that ...
2
votes
2answers
72 views

Differentiability of a certain piecewise function

Consider the function $$ f(x)=\begin{cases} x & \textrm{if } x \textrm{ is rational} \\ -x & \textrm{if } x \textrm{ is irrational} \end{cases} $$ It is well-known that $f(x)$ is continuous at ...
8
votes
2answers
69 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
2
votes
1answer
52 views

Does there exist a suitable function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(n^m) = mf(n)$? I think no.

Any ideas how to prove that no injection $f : \mathbb{N} \rightarrow \mathbb{N}$ whose image is closed under multiplication by elements of $\mathbb{N}$ satisfies the following identity? $$f(n^m) = ...
3
votes
1answer
173 views

Counterexample for a non-measurable function?

I am struggling to solve an exercise in my measure theory book and any help for solving it would be appreciated: Let $(\Omega,\mathcal{A},\mu)$ be a measure space and let $f:\Omega \to \mathbb{R}$ ...
2
votes
9answers
5k views

Example of functions that are onto but not one-to-one

I have been preparing for my exam tomorrow and I just can't think of a function that is onto but not one-to-one. I know an absolute function isn't one-to-one or onto. And an example of a one-to-one ...
1
vote
2answers
138 views

Function oscillating between $[-1,1]$ around $0$.

I was looking for examples of functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $a>0$ we have that $[-1,1]\subseteq f\left([-a,a]\right)$ The two examples I could think of were $\sin(1/x)$ ...
1
vote
1answer
84 views

Looking for a counterexample [duplicate]

Possible Duplicate: Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…) I am looking for a function $f:\mathbb{R}\rightarrow \mathbb{R}$ that for all $x$ and $y$ ...
4
votes
2answers
193 views

Existence of an infinitely differentiable function $ f $ with $ {f^{(n)}}(0) = 0 $ for all $ n \in \mathbb{N} $.

How can one show that there exists an infinitely differentiable function $ f: \mathbb{R} \to \mathbb{R} $ such that $ {f^{(n)}}(0) = 0 $ but $ f^{(n)} \not\equiv 0 $ for all $ n \in \mathbb{N} $?
5
votes
3answers
317 views

Does the Laplace transform biject?

Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.' Can someone provide a proof or counterexample ...
4
votes
4answers
275 views

Examples of partial functions outside recursive function theory?

My math background is very narrow. I've mostly read logic, recursive function theory, and set theory. In recursive function theory one studies partial functions on the set of natural numbers. Are ...
7
votes
2answers
312 views

Must $g$ be the identity if $f = g \circ f$?

I am finding it hard to solve the following problem. Let $A$ is a set and $f : A \rightarrow A$ and $g : A \rightarrow A$. If $f = g \circ f$, must $g$ be an identity function always? Will there be ...
5
votes
1answer
1k views

Construct a monotone function which has countably many discontinuities

I read in a textbook, which had seemed to have other dubious errors, that one may construct a monotone function with discontinuities at every point in a countable set $C \subset [a,b]$ by enumerating ...
4
votes
3answers
680 views

non time constructible functions

A function $T: \mathbb N \rightarrow \mathbb N$ is time constructible if $T(n) \geq n$ and there is a $TM$ $M$ that computes the function $x \mapsto \llcorner T(\vert x\vert) \lrcorner$ in ...
13
votes
4answers
467 views

Example of a rational function such that : $(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$

Can any one give me example of: rational functions $f, g$ and $h$ with rational coefficients such that $$(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$$ Also, if anyone knows a procedure for constructing ...