4
votes
1answer
72 views

An explicit construction for an everywhere discontinuous real function with $F((a+b)/2)\leq(F(a)+F(b))/2$?

I would like to know an explicit method on constructing an everywhere discontinuous real function $F$ with the property: $$F((a+b)/2)\leq(F(a)+F(b))/2.$$ There is a non-constructive example (with the ...
1
vote
4answers
77 views

How do I tell whether axiom of choice is used or not?

I am having a hard time understanding the Axiom of Choice(AC). Say I have an index set $A$ , and a collection of indexed sets {${V_\alpha}$}, where $\alpha$ is a member of $A$. Then, does the ...
0
votes
2answers
180 views

Why is the l.u.b. property equivalent to Cauchy-sequence convergence for $\mathbb{R}$?

Math people: I browsed some questions with similar titles and could not find a duplicate. I apologize if it is. If you read my question it will be obvious that I am not a logician, so please be ...
10
votes
1answer
254 views

Does a nonlinear additive function on R imply a Hamel basis of R?

A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
0
votes
1answer
77 views

Analysis Proof without Axiom of Choice

Lemma: Given any function $f:M\to N$ where $M$ and $N$ are both metric spaces, $\lim_{x\to a}f(x)$ converges to $L$ only if given any function $\gamma:\mathbb{I}\to M$ (where $\mathbb{I}$ is the unit ...
5
votes
1answer
137 views

Does the specification of a general sequence require the Axiom of Choice?

Many results in elementary analysis require some form of the Axiom of Choice (often weaker forms, such as countable or dependent). My question is a bit more specific, regarding sequences. For ...
7
votes
1answer
304 views

Continuous root map of the coefficients of a polynomial

I have a set of polynomials $P_t(z)= z^n+ a_{n-1}(t)z^{n-1}+\cdots+ a_0(t)$ which depends on a real parameter $t \in [a,b]$ and where $a_{n-1}(t),\ldots, a_0(t)$ are real continuous functions. May I ...
18
votes
2answers
1k views

Continuity and the Axiom of Choice

In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ ...
1
vote
2answers
632 views

Compact but not sequentially compact question

At this page: http://planetmath.org/encyclopedia/SequentiallyCompact.html you can find an example of a compact but not sequentially compact space. My question is: how to prove the existence of "$r ...
3
votes
2answers
246 views

Question about a puzzle on injecting a subset of $\mathbb{R}$ into $\mathbb{Q}$

I was just browsing through the Puzzle section on Noam Elkies website. The puzzle can be found here. The solution to the puzzle proves that any well-ordered subset of $\mathbb{R}$ is countable. In ...