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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

33 votes
Accepted

Maximal ideals in the ring of continuous real-valued functions on ℝ

Peter Johnstone's book Stone Spaces (p. 144) proves that for any X, maximal ideals in $C(X)$ are the same as maximal ideals in $C_b(X)$ (bounded functions), i.e. the Stone-Cech compactification $\beta …
Martin Sleziak's user avatar
2 votes

Complement of a finite union of convex sets

This follows from Alexander duality, and works with $\mathbb{R}^2$ replaced by $\mathbb{R}^d$ for any $d$. In more detail, let us compactify $\mathbb{R}^d$ to $S^d$ and consider $F'=S^d\setminus\bigc …
Eric Wofsey's user avatar
  • 31.2k
25 votes
Accepted

Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Any map $[0,1]\to[0,1]$ is a quotient map onto its image, and the image must be either a point or a closed interval. So a partition comes from such a map iff the quotient of $[0,1]$ by the equivalenc …
Eric Wofsey's user avatar
  • 31.2k
4 votes

Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non...

It's not hard to construct a compact connected LOTS with this property. Let $X_0$ be any countably saturated dense linear order, and let $X$ be its bounded Dedekind completion ("bounded" meaning also …
Eric Wofsey's user avatar
  • 31.2k
12 votes
Accepted

Are all well behaved "mean" functions on $\mathbb{R}^+$ equivalent?

Define a mean algebra to be a set $S$ with an binary operation $M$ satisfying (1), (2), and (4). We can define $M(a,b,c,d)=M(M(a,b),M(c,d))$ and this will depend only on the multiset $\{a,b,c,d\}$. …
Eric Wofsey's user avatar
  • 31.2k
17 votes

Continuous relations?

Here is an expansion of my comment into an answer which I think is very compelling as the "correct" definition for compact Hausdorff spaces, though I agree with others who have said that for general s …
Lehs's user avatar
  • 862
5 votes

The role of the mean value theorem (MVT) in first-year calculus

I have just been teaching calculus for the first time, and I am firmly of the opinion that in many calculus courses, the mean value theorem should have essentially no role. All the applications of it …
Eric Wofsey's user avatar
  • 31.2k
28 votes

Does pointwise convergence imply uniform convergence on a large subset?

A simple diagonalization argument gives a counterexample if you assume the continuum hypothesis. Let a modulus of convergence be a sequence $\delta:{\mathbb N}\to(0,\infty)$ which converges to 0, and …
Eric Wofsey's user avatar
  • 31.2k