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Results tagged with real-analysis
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user 5903
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
8
votes
Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
You can actually make $S$ a (copy of the) Cantor set.
First a claim: if $[a_1,b_1]$, $[a_2,b_2]$, ..., $[a_n,b_n]$ is a finite set of intervals, ordered left-to-right ($b_1<a_2$, $b_2<a_3$, etc) then …
- 11.4k
7
votes
On the definition of a continuous function
The function $f$, defined by $f(0)=0$ and $f(x)=\sin\frac1x$ if $x\neq0$, is not continuous at $0$, but its graph is connected over every open interval that contains $0$.
With a bit of work, shifting …
- 11.4k
1
vote
Condition for set of the type $\{(a,b)|a \in A, \ b = f(a)\}$ to have empty interior if $A$ ...
If the interior of $X$ were nonempty there would be nonempty open sets $U$ and $V$ in $L^1$ such that $U\times V\subseteq X$. But then $U\subseteq A$ shows that $A$ would have nonempty interior.
Note …
- 11.4k
1
vote
Direct proof a property of hyperstonean spaces
I take it that the normal, self-adjoint functions are assumed to be continuous on their domains.
In that case the functions $f+g$ and $f\cdot g$ are continuous on the dense open set $O=X\setminus(Z_f\ …
- 11.4k