# Tagged Questions

1answer
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### 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $z_n \to a$ ...
2answers
632 views