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) ...

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function nondecreasing in both variables, set of discontinuities is a nullset

Let $f\colon [0,1]^2\to\mathbb{R}$ be a function such that $g(x):=f(x,y)$ for any $y$ and $h(y):=f(x,y)$ for any $x$ are nondecreasing functions (the second variable is fixed). Prove that the set of ...
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a continuous function on $\mathbb{Q}$

Is there a continuous bijective function from $[0,1] \cap \mathbb{Q}$ to $\mathbb{R}$? I think that there is no such function. The set $|[0,1] \cap \mathbb{Q}|$ is countable and $|\mathbb{R}|$ is ...
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Is Inverse of a function continuous too?

I read an example from "Principles of Mathematical Analysis" by Rudin under the section 'Continuity and Compactness'. According to the example, Let $X$ be the half-open interval $[0,2\pi)$ on the ...
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If $f:[a,b]\rightarrow R$ is a uniformly continuous function then its absolutely continuous?

If $f:[a,b]\rightarrow R$ is a uniformly continuous function then is it true that $f$ is always absolutely continuous?
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What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
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Geometric generation principle form constructing the Hilbert Curve

I have some questions on the generation of the Hilbert's space-filling curve. Any help to clarify doubts a-e would be very appreciated. The Hilbert's space-filling curve is a function $f_h:[0,1]\...
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50 views

Are eigenvalues (resp. unit eigenvectors) dependent continuously on elements $a_{ij}$ of a symmetric matrix $A$? [closed]

Let $A(t)=(a_{ij}(t)),~(t\in \mathbb R)$ is a symmetric matrix such that $a_{ij}(t)=a_{ji}(t)$ is a real-valued continuous function. Let $\lambda_1(t) \ge \cdots \ge \lambda_n(t)$ is all of the ...
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Continuity of Holder functions

If a function taking values in $\mathbb{R}^n$ is $\alpha$-Holder continuous along lines parallel to the axes (uniformly on a compact set), is it continuous?
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Prove that $f(x)=\begin{cases} \frac{x}{x-4}, & x\not= 4 \\ 0, & x=4 \end{cases}$ is continuous.

Prove that the function $f(x)$ defined by $$ f(x)=\begin{cases} \dfrac{x}{x-4}, & x\not= 4 \\ 0, & x=4 \end{cases} $$ is continuous. My question is: Do I have to prove the two sides ...
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$\mathcal{f}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ restricted to sections is continuous implies continuity

Let $\mathcal{f}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that $\mathcal{f}$ restricted to each {$x=a$} is continuous and restricted to each section {$y=b$} is continuous and monotone.Prove that $\...
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For a linear function the following are equivalent: continuity and Lipschitz continuity

Let $(X,||\cdot ||_X)$ and $(Y,||\cdot ||_Y)$ be normed Vectorspaces over a common field $\Bbb K$. Let $A:X \to Y$ be a linear function. I have to show that the following statements are equivalent: $...
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Continuity of Lipchitz constant of local lipschitz function

Suppose $f:\mathbb{R}\to \mathbb{R}$ be local lipschitz, which is equivalent to Lipschitz on compact sets. That is, for any $R>0$, there exists some $L >0$ such that $$\sup_{|x|,|y|\le R}\...
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Whether the function $f(x,y)$ is continuous at $(0,0)$

QUESTION: $$f(x,y)=\begin{cases}x \sin \frac{1}{y} + y \sin \frac{1}{x} & \text{if } xy \not = 0 \\ 0 & \text{if } xy = 0\end{cases}$$ Show that $f(x,y)$ is continuous at $(0,0)$. ...
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Infinite differentiability of a function with a removable discontinuity

How would I prove that $\frac x{e^x-1}$ is infinitely differentiable? (This question came up since the No 1 answer in Maclaurin series for $\frac{x}{e^x-1}$ states that the function is infinitely ...
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Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$

$\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | ); \: \: \: \: \: \: \phi(u):=\int_0^1 u^2(t) dt $ Is this function continuous or even uniformly continuous? (I know that the ...
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Proof that function on topological space is continuous if and only if 2 restrictions of it are

Topology such that function is continuous if and only if the restriction is. I've already seen this post but it didn't really help. The problem is the following: Let $X$ and $Y$ be topological ...
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Testing differentiability and continuity

Consider the following function $ f(x) = 0 $ if x is rational $ f(x) = x^2$ if x is irrational Then only one of the following statements is true which one is it ? a.) $f$ is differentiable at $...
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Lipschitz-continuity of $x\mapsto\frac{x}{||x||}$ in a general Banach space

Let $(X,||.||)$ be a Banach space. Assume we have constants $0<C_1<C_2<\infty$. Define the set $A:=\{x\in X\text{ }|\text{ } C_1\le ||x||\le C_2\}$. Is the map $f\colon A\rightarrow X$, $x\...
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$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
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Differentiablity at $0$ of a function $f: \mathbb R \to \mathbb R$ which is twice differentiable in $\mathbb R \setminus \{0\}$

Let $f: \mathbb R \to \mathbb R$ be a function , twice differentiable in $\mathbb R \setminus \{0\}$ such that $f'(x)<0<f''(x) , \forall x <0$ and $f'(x)>0>f''(x) , \forall x >0$ ; ...
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Continuous or Differentiable but Nowhere Lipschitz Continuous Function

What is a real valued function that is continuous on a close interval but not Lipschitz continuous on any subinterval? What is a real valued function that is differentiable on a close interval but not ...
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Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?

Here is Prob. 20 (a) in the book Topology by James R. Munkres, 2nd edition. Consider the product, uniform, and box topologies on $\mathbb{R}^\omega$. In which of these topologies are the ...
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If a linear map $T:X^*\to X^*$ is norm-norm continuous, is it weak-star - weak-star continuous?

Let $X$ be a Banach space and suppose $T:X^*\to X^*$ is a linear mapping. If $T$ is norm-norm continuous, i.e. continuous from the normed space $X^*$ into the normed space $X^*$, is it also continuous ...
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How does this faulty system of integration change the nature of jump discontinuity?

Let's define a sort of faulty integral. For the purposes of this question we shall assume that this is the regular integral. This integral integrates all functions properly however it's gets confused ...
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deformation retraction as mapping cylinder

In Hatcher's Algebraic Topology, the mapping cylinder is defined as the quotient space of the disjoint union $(X\times I)\sqcup Y$ (where $I$ is the unit interval) of a continuous $f:X\to Y$, where $(...
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Continuous function math question

This is the last question I have to answer for my math class. I thought I understood the concept of a continuous function, but I can't seem to get this one right. I only have 1 more submission attempt,...
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Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism [closed]

Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism Requirements for a homeomorphism $f:X \rightarrow Y$: $f$ is ...
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$f$ continuous $\iff f(B(a,\delta))\subset B(f(a),\epsilon)$

My book says that when $f$ is continuous, we have that $\forall \epsilon>0$, there exists $\delta>0$ such that: $d(x,a)<\delta \implies d(f(x),f(a))<\epsilon$ Then, my book says that ...
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$\partial A$, when $A=\{x\in M: f(x)>0\}$ is the set $\{x\in M: f(x) = 0\}$

I have a question about the proof of this fact: $\partial A$, when $A=\{x\in M: f(x)>0\}$ is the set $\{x\in M: f(x) = 0\}$ The proof says the following: $$A = f^{-1}((0,+\infty))$$ Since $(0,+\...
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$f \in C(\mathbb R)$ such that $f\Big(x+\dfrac 1n \Big) > f\Big(x-\dfrac 1n \Big) , \forall x \in \mathbb R , n \in \mathbb N$ ; is $f$ increasing?

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $f\Big(x+\dfrac 1n \Big) > f\Big(x-\dfrac 1n \Big) , \forall x \in \mathbb R , n \in \mathbb N$ , then is it true that $f$ is ...
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For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N

In order to prove: For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N I'm supposing that $x_n$ is convergent, that is: $$\forall \epsilon>0, \...
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How do I examine the function on continuity? How do I discuss and sketch the level lines of f?

How do I examine the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, $f (x, y) = (2x- y)\ \rm{sign}(4x-y)^2$ for continuity? How do I discuss and sketch the level lines of $f$?
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Real Analysis question on FTC, Integral

Let $g:[0,1] \rightarrow \mathbb R$ be a continuous function and assume that $$ \int_{0}^{1} g(x) \phi'(x) dx = 0 $$ for all continuously differentiable functions $\phi: [0,1] \rightarrow \mathbb ...
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Simple discontinuity of real valued function

Consider a real valued function $f$ defined on $[a,b]$. Say it has a simple discontinuity at a point $x \in (a,b)$, with $f(x-) \neq f(x+)$ (LHL not equal to RHL). Is it necessary that $f(x)$ is ...
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Are slices $\left\{b\right\}\times F\subset B\times F$ homeomorphic to $F$? [closed]

Looking at a continuous projection $B\times F\rightarrow B$, are slices $\left\{b\right\}\times F\subset B\times F$ homeomorphic to $F$?
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Graph of a continuous function $f:M\to N$ is a closed subset of $M\times N$

I need to prove that the graph of a continuous function $f:M\to N$ is a closed subset of $M\times N$. $N$ is a metric space. I think I'm supposed to use this result. So, that's what I did: $Graph(f) =...
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$M=A\cup B$, $f|_A$ and $f|_B$ are continuous, then $f$ is continuous in $A\cap B$

In order to prove: $M=A\cup B$, $f|_A$ and $f|_B$ are continuous, then $f$ is continuous in $A\cap B$ does it suffice to prove: for $a\in A\cap B$: since $f|_A$ is continuous, then $\forall \...
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Prove $g(y) = \int_{\mathbb{R}} \sin(y^2x)f(x) dx$ is bounded and continuous on $\mathbb{R}$ for $f \in L^1(\mathbb{R})$

This question is from a practice qualifying exam. Here's my attempt (I'm a bit stuck on the continuity part): Since $f \in L^1(\mathbb{R})$, $f$ is bounded. Then: $$|g(y)| = |\int_{\mathbb{R}} \sin(...
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Continuity in terms of interior of preimage and preimage of interior

Let $f$ be a map between metrix spaces $X,Y$. In order to prove: $f$ is continuous $\iff$ $f^{-1}(\operatorname{Int} Y)\subset \operatorname{Int}(f^{-1}(Y))$ I did: $\rightarrow$ Suppose $x\in f^{-1}...
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How to find the set of values $S$ where $f$ is not differentiable?

Let's assume we are given an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and for the purposes of this question, let's assume we know nothing about the differentiability of $f$, i.e. we have no ...
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Proving equivalence of statements on continuity between metric spaces

On page 228 of Mícheál Ó Searcóid's Metric Spaces (2007), he writes Criteria for Comparability of Metrics Suppose $X$ is a set and $d$ and $e$ are metrics on $X$. Then the following ...
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Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a "...
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$t\mapsto\sin(tA)$ is continuous

How to show that $t\mapsto\sin(tA)$ is continuous for a real matrix $A\in Mat(n,n)$ Can I use trigonometric identity, $\sin y-\sin x=2\cos\left(\frac{x+y}{2}\right)\sin(y-x)$ but this holds only ...
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126 views

Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the Spectrum$(S^...
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Can you use level sets to suggest discontinuity?

Consider the function $$f(x, y) = \frac{x^2 + y^2}{y}$$ for which I already showed that the level set of height $c$ is given by a circle of the form $$C_c: (x - 0)^2 + (y - \frac{c}{2})^2 = \frac{c^...
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1answer
14 views

Removing Discontinuity in 3-space without changing the partial derivative

Is it possible to find a version of the function $$f(x,y) = x\cdot \lfloor y \rfloor + \lfloor x\rfloor^2$$ That is continuous. ANY operation is allowed in changing the function as long as the ...
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40 views

Simple example of a mapping between topological spaces

I read the definition of a continuous function between topological spaces a lot of times, but I'm having difficulties to apply it to a simple example. Given two topological spaces $(X,\tau_1)$ and $(...
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125 views

On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C$, $D$ are the rings of continuous, respectively differentiable ...
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31 views

Obtaining a bound on the Bernstein approximation of Lipschitz functions

I encountered the following excercise in a book: Exercise: Given a Lipschitz continuous function $f$ on $[0,1]$, with Lipschitz constant $c$. Show that $|B_{n,f}(p) - f(p)|\le \frac{c}{2\...
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Continuous indicator-like functions

Let $\Omega$ be a compact subset of $\mathbb{R}^n$. Let $g:x\in\mathbb{R}^n\to\mathbb{R}$ be a continuously differentiable function such that $$ \begin{cases} g(x)>0 & x\in\text{int}\Omega,\\ ...