All Questions
Tagged with discontinuous-functions or continuity
17,686 questions
0
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Getting different answers for integration problem: $\int_0^2 x d( \{x\} )$
My teacher used integration by parts to solve the problem like so:
$$\int_0^2 xd(\{x\})
=[x\{x\}]_0^2-\int_0^2 \{x\}dx\\
=0-\int_0^1 xdx-\int_1^2 (x-1)dx$$
which comes out to -1. But when I was ...
2
votes
0
answers
90
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Prove: If $f:\Bbb{Z}\to\Bbb{Z}$ and along divisor chains $n_1\mid n_2\mid\dots\implies f(n_1)\geq f(n_2)\geq\dots$, then $f$ has profinite continuity
Let $\sum_{d\mid n}\cdot$ be a sum over all positive divisors of $n$. Notice what happens when you take $n = 0$, the sum range can be over all $d\in \Bbb{N}$, which is an infinite number of terms, ...
- 22.7k
-1
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1
answer
46
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How to prove that $C^k(\mathbb{R})$ is a subspace of $F(\mathbb{R};\mathbb{R})$?
I'm starting my linear algebra studies and came across the following statemtent:
$E = F(\mathbb{R};\mathbb{R})$ is the vector space of the one variable real functions $f:\mathbb{R} \rightarrow \...
11
votes
3
answers
2k
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Can there be a continuous function with infinite derivative everywhere?
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. For all nowhere-differentiable examples that I know of, for each $a\in\mathbb{R}$ there exist sequences $x_n\to a$ and $y_n\to a$ such that
$$\frac{f(...
- 9,329
4
votes
1
answer
140
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Does there exist a function such that $f(x), f(x)+\sqrt{3}, \sqrt{2}-f(x), f(x)+x$ are irrational for all irrational $x$?
Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(x), f(x)+\sqrt{3}, \sqrt{2}-f(x), f(x)+x$ are irrational for all irrational $x$?
My attempt: I couldn't come up with any good ...
- 593
1
vote
1
answer
79
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Topology literature focusing on continuous functions and measures
I am looking for literature dealing with topologies on spaces of continuous functions ($C_0$, $C_c$, $C_b$, $\ldots$), particularly with regard to their application when dealing with topologies on ...
- 102
0
votes
1
answer
28
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Continuity of piecewise application between Metric Spaces
I have the following exercise and I don't know if my proof is correct:
Let $(X,d_{X}),(Y,d_{Y})$ metric spaces with $X=X_{1}\cup X_{2}$
Let $f_{i}:X_{i}\to Y, \ i\in\{1,2\}$ continuous applications ...
- 337
1
vote
0
answers
49
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Since rings of continuous maps exist why in (co)homology do you never see chain complex $C_n =$ certain ring of continuous functions "of degree $n$"?
I'm new to homological algebra. Just wondering why we never seem to see the involved chain complexes defined simply to be $C_n=$ continuous maps "of degree $n$" where the context makes the ...
- 22.7k
1
vote
1
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108
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$f$ is continuous iff $f_{1},\dots,f_{n}$ continuous in Metric Spaces
I`m trying to do the following exercise for my general topology class:
Let $(X,d)$ a metric space and $B\subset\mathbb{R}^n$ with euclidean
metric. Let $f:X\to B$ and application determined by $f_{i}...
- 337
0
votes
2
answers
49
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Real function is continuous iff image of closure is subset of closure of image [duplicate]
Let $f:\mathbb{R}\mapsto \mathbb{R}$. The goal is to prove that $f$ is continuous if, and only if, for all $X\subset \mathbb{R}$, $f(\overline{X})\subset \overline{f(X)}$
Let $X\subset \mathbb{R}$, $y\...
- 357
4
votes
2
answers
131
views
continuity in Munkres Topology Lemma 54.1
Lemma 54.1 of Munkres states a unique lift $\bar{f}$ of a path $f$ through a covering map $p$. In the proof of this lemma, he constructed the lift step by step. He assumed $\bar{f}$ is defined in the ...
- 2,613
5
votes
0
answers
156
views
Topological obstructions to finding a map $g\colon \mathbb{R}^2 \longrightarrow S^2$ subject to constraints
Given a continuous map $f\colon \mathbb{R}^2 \longrightarrow S^2$, is it possible to find a continuous map $g\colon \mathbb{R}^2 \longrightarrow S^2$ such that
$g(x) \neq f(x)$ for all $x \in \mathbb{...
- 299
1
vote
1
answer
28
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Well defindness of homotopy map $F(s,t)=h\left(f(s-st)\right)$
Let $I=[0,1]$, $h:(X,x_0) \to (Y,y_0)$ and $[f] \in \pi_1(X_0)$. Can I construct the following map $$F(s,t)=h\left(f(s-st)\right):I \times I \to Y?$$
Here, $F(s,0)=h \circ f(s)$ and $F(s,1)=e_{y_0}$. ...
- 2,613
2
votes
0
answers
115
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Does the existence of a continuous bijection give a useful preorder on the class of topological spaces?
Consider the binary relation $\gtrsim$ between topological spaces defined by
$A \gtrsim B$ iff there exists a continuous (not necessarily bicontinuous!) bijection from $A$ to $B$.
This relation is ...
- 6,950
3
votes
2
answers
360
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Does there exist a continuous surjection between any two compact (or any two non-compact) manifolds?
This answer says (rewording slightly for clarity)
There exists a continuous surjective map from ... $\mathbb{R}^k$ for any $k≥1$ to [any] (separable) connected, $n$-dimensional topological manifold. ....
- 6,950