Questions tagged [continuity]
Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)
17,686 questions
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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 ...
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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, ...
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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 \...
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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(...
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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 ...
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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 ...
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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 ...
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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 ...
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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}...
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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\...
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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 ...
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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{...
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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 $F(s,1)=e_{y_0}$. ...
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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 ...
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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. ....
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