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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Oscillation of function and continuity clarification of proof

There is a similar question related to this, but it doesn't answer my question, so I would be thankful if anyone helped me with it. There is a step specifically in the proof I do not understand. ...
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Need help to prove this function

Suppose that $f:R\to$ $R$ is continuous on $R$, and that $f(r)=0$ for every rational number r. Prove that $f(x)=0$ for all $x$ in $R$. I let r in $R-Q$, and $f(x)= lim(f(r_n))=f(lim(r_n))=lim(0)=0$. ...
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There is a bijection between $\{ f: X \to Y \text{ continuous} \}$ and $ \tau_{_X} \times \tau _{_Y}$

Let $\tau _{_X}$ and $\tau _{_Y}$ denote the topologies on $X$ and $Y$ respectively. I know that the statement in the topic is not true, but my feelings say that with a small modification, this will ...
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Wouldn't any arbitrary natural number work to to bound $|x-x_0|$ (example of showing a function is contious)

Here is an excerpt from Ross' Elementary Analysis where he shows $f(x)=2x^2+1$ is continuous using the $\epsilon$-$\delta$ definition. Couldn't have Ross used any natural number when he wanted to ...
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Translation operator and continuity

I came across a text that proves that translation operator $T_a(f):=f(x-a)$ where $a\in\mathbb{R}^n$ and $f\in L^p(\mathbb{R}^n)$ is continuous. The proof follows: ...
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Extending a function continuously

Let $f:\mathbb{R}\to \mathbb{R}$ be continuous in an interval around some real $c\in (a,b)$. Construct $g:\Bbb{Q}\setminus \{0\}\to \mathbb{R}$ as : $$g(s):=\frac{f(c+s)-f(c)}{s}\quad s\in ...
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About continuous functions and continuous continuations and their uniqueness

How would you access the following problem: (a) Show that for every $s \in \mathbb{Q}$ the function $$f: \mathbb{C}^* \rightarrow \mathbb{C}$$ $$ f(z) := \frac{\overline z}{\vert z \vert ^s}$$ is ...
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Finding an unbounded set with a specific property

Find an unbounded subset $A ⊂ \Bbb R$ such that every function from $A$ to a metric space is uniformly continuous. My attempt at the solution (incomplete). If $A⊆ \Bbb R$ were such a set, then for ...
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If $f$ is continuous on $[0,\infty)$ and not bounded above implies…

$Conj:$ If $f$ is continuous on $[0,\infty)$ and not bounded above implies there exists $\{ x_n\}$ such that $x_n \rightarrow \infty$ and $f(x_n)\rightarrow \infty$. I can see how to show $f(x_n) ...
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An Application of Intermediate Value Theorem

Let $f :\Bbb R→ \Bbb R$ be given by $f(x) := x^{n}$ for some $n ∈ \Bbb N$. If $b$ is a positive real number, show that there exists a unique positive real number $a$ such that $a^{n} = b$. My ...
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How to show that $F(x)$ continuous?

$F:\Big(C[0,1],||.||_2\Big)\rightarrow \Big(C[0,1],||.||_3\Big)$ $x\rightarrow F(x)(t)=\int^t_0x(s)ds,\quad\quad0 \le t\le 1 $ Show that F is continuous. F is linear. for n=0,1,2.. $x_n(t)=t^n,0 ...
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50 views

Real analysis. Uniformly continuous

Suppose $$f:\mathbb R\to\mathbb C$$ is continuous and $f(x)=0$ for all $|x|>1$. Show $f$ is uniformly continuous on $\mathbb R$. This is not homework. I'm trying to study for a test. I ...
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Infinitely many times differentiable function with unbounded derivatives?

Let $f$ be an infinitely many times continuously differentiable function on the compact interval $[0,1]$. We denote by $f^{(k)}$ the $k$-th derivative with respect to $x$. Then we know: $\sup_{x \in ...
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29 views

An obstacle in a proof of Lagrange's mean value theorem by Nested Interval theorem

I was trying to prove Lagrange's mean value theorem by Nested Interval theorem and there's step where I got stuck ; let me write down to the step Let $f:[x_1,x_2]\to \mathbb R$ be continuous on ...
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Proving uniform continuity using limits

Hi I am interested in a result which states that if a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ has finite limits on both sides then the function is uniformly continuous. Is it ...
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Function for which it is unknown whether it is continuous

Is there any function $f:\mathbb R\rightarrow \mathbb R$ for which at least some values are known but it is unknown whether $f$ is continuous or not? Edit: I am looking for examples from actual ...
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16 views

Hints required on a problem of continuous function

Let $f:\mathbb R\rightarrow \mathbb R$ be a function with intermediate value property.Let $x\in \mathbb R$ .Suppose to each sequence $x_n\rightarrow x \exists $a constant $M$ such that ...
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Why does this specific $\delta$ imply a failure of $|f(x)-f(x_0)|< \epsilon$? (trouble in understanding continuity proof)

I'm having a bit of trouble with a the proof in Ross' Elementary Analysis. The theorem is the $\epsilon-\delta$ one. Theorem: Let $f$ be continuous at $x_0$ in $dom(f)$ if and only if for each ...
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39 views

Every function from discrete metric space to another metric space is uniformly continuous

My solution:It is fairly straightforward graphically but I just want to ensure if it is rigorous enough. Suppose $X$ is a discrete metric space and $f$ be any function from $X$ to $Y$ where $Y$ is ...
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proving C2 continuity given closed spline constraints

Given the closed spline's constraints as below $$P(0) = P_k $$ $$P(1) = P_{k+1}$$ $$P''(0) = P_{k-1} - 2P_{k}+P_{k+1}$$ $$P''(1) = P_{k} - 2P_{k+1}+P_{k+2}$$ How do I prove that this spline ...
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Question regarding the sequence definition of continuity.

Here is an excerpt from Ross' Elementary Analysis (specifically the definition of continuity): "The function $f$ is $\it \space continuous\space at \space x_0$ if, for every sequence $(x_n)$ in ...
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A closed, bounded subset $A$ of $\Bbb Q$ and a continuous function $f : A → \Bbb R$ such that $f$ is not bounded

Find a closed, bounded subset $A$ of $\Bbb Q$ and a continuous function $f : A →\Bbb R$ such that $f$ is not bounded Note: $\Bbb Q$ is the set of all rationals. My Solution: $A=\{x:x\in\Bbb Q, ...
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A continuous path between shapes

Let $A$ and $B$ be two geometric shapes in the plane (two measureable subsets of $\mathbb{R}^2$) such that $A\subseteq B$. Define a $path$ from $A$ to $B$ as a function $f$ from $[0,1]$ to subsets of ...
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30 views

Continuity of $\sup_{x\in\Omega}\varphi(x,\cdot)$

Let $\Omega\subset\mathbb{R}^n$ be open,bounded and (I don't know if this matter) of class $C^{1+\alpha}$. Let $\varphi:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $\varphi(x,\cdot)$ is ...
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Prove sin(1/x) is discontinuous at 0 using epsilon delta definition of continuity

Let $$f(x) = \begin{cases} 0 &\text{ if $x=0$}\\ \sin(1/x) &\text{ otherwise} \end{cases} $$ Prove that $f$ is discontinuous at $0$ using the $\epsilon \delta$ definition of continuity. I ...
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Lipschitz continuity of a function

Show that $g(x) = \frac{1}{x^{2} +1}$ is Lipschitz conitnuous. From the definition, we must show that $\forall x,y \in \mathbb{R}$, $|f(x)-f(y)| \leq K|x-y|$, for some real constant $K$. First, I ...
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If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .

Exercise: Suppose that $f$ is continuous on $[a,b]$ and that $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ . Attempt of proof: Suppose that $f$ is continuous on $[a,b]$ and that ...
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Show that the function is discontinuous at c=2 using delta epsilon

I am struggling with discontinuity and continuity. f(x)= {-1, x<2 0, x=2 1, x>2 I realize that I must show that $\exists$ $\epsilon$ > 0 ...
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Prove continuity of $\frac{xy}{x^2 + y^2}$ using formal definition

I need to prove that the function $f(x, y) = \frac{xy}{x^2 + y^2}$ is continuous on $(x,y) \in \mathbb{R}^2 - (0,0)$ using the following definition of continuity: Let S be a subset of $\mathbb{R}^p$. ...
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How do we 'know' that $2^x$ is continuous?

It is intuitive for $2^n$, if $n$ is an integer, to exist. How do we know that less intuitive values such as $2^\frac{1}{2}$, $2^\sqrt{2}$, $2^\pi$ etc exist? I'd like to accept that $2^x$ is ...
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Continuity of $e^{-x^2/4k}$

I want to show that $f(x)=e^{-x^2/4k}$ (where $k>0$ is fixed) is continuous using an $\epsilon$, $\delta$ argument. I've been trying to choose $\delta$ using $\ln$ somehow and I've also been trying ...
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Continuity from property of constriction images of spheres

Let $D\subset\mathbb R^n$ --- domain and mapping $\varphi:D\to \mathbb R^n$. The following property holds There is a set $T\subset D$ s. t. measure $|D\setminus T|=0$ and for every point ...
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Mulltivariable limit quesion and my attempt

Given $$ F(x,y) = \left\{ \begin{array}{ll} 0 & (x,y)=(2y,y) \\ \exp \biggl( \frac{|x-2y|}{x^2 -4xy +4y^2} \biggr) & (x, y) \ne (2y, y) \end{array} \right. $$ Task is to examine whether ...
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$\epsilon-\delta$ continuity definition domain

Does epsilon-delta continuity implicitly requires that there would be at least one non-trivial Cauchy sequence converging in the function's domain? Generally the criteria is introduced with no ...
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right-continuous function with left limit existing compact image

Consider a right-continuous function with left limit existing, what can we tell about its image on compact set ?
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Continuity and differentiability of two variables function

Let be $f:\mathbb{R^2}\rightarrow\mathbb{R}$ defined by: $$f(x,y)= \begin{cases} x^3\log{\left(1+\frac{|y|^\alpha}{x^4}\right)} & \text{if } x \neq 0 \\ 0 & \text{if } x =0 \end{cases}$$ ...
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Extending a continuous function to the closure

I'm dealing with the following problem: Let $X$ a topological space, $Y$ a metric space and $A$ a subspace of $X$. If $f$ is a continuous mapping of $A$ into $Y$, show that $f$ can be extended in ...
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trying to prove: If $f$ is continious and is lebesgue-almost-everywhere constant, then it is constant

I was wondering if this claim is true, and if it is then how would one try to prove it: If $f\in C[0,1]$ (and thus is continuous) and is Lebesgue-almost-everywhere constant, then it is constant. It ...
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semi continious functions characterizations

Does anyone knows how to prove this: Let $f: (X, d) \rightarrow \mathbb{R}$ be an upper semi-continious function. Prove that $f$ is u.s.c. if and only if $ \{ x \ \ |\ \ f(x) \geq z \} $ is closed ...
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Show there is an unbounded continuous function on a closed nonempty subset of a metric space.

Let $X$ be a metric space. Let $E \subset X$ be not closed and nonempty. Show that there is a continuous real-valued function on $E$ that is not bounded. The only function that I know of that is ...
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Continuity in argument of minimization

Let $$g(c) = \min_{Ax=c} f(x),$$ where $x$, $c$ are vector-valued, $A$ is a matrix and $f$ is a smooth convex function. Under what conditions can we say $g(c)$ is continuous in $c$?
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Principal bundle map is fiber homeomorphism

let $B_1(\mathcal{P}_1:P_1\rightarrow X_1)$ and $B_2$ be two principal G-bundles and let $\tilde f:P_1 \rightarrow P_2$ be a principal bundle map. I want to prove that $\tilde f$ carries each fiber of ...
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Existence of Limit iff $x',x'' > X, |f(x')-f(x'')| < \epsilon$

I was given a theorem in class regarding uniform continuity that does not appear in my textbook. It says that $$\lim_{x \to \infty} f(x) = a \iff \text{ for all } x',x'' > X, |f(x')-f(x'')| ...
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How to use the Mean Value Theorem to show the continuity of a case-defined function?

Let $$f(x)=\frac{e^x-1}{x}, \quad x\neq 0$$ $$f(x)=1, \quad x=0$$ Use the Mean Value Theorem to show that $f$ is continuous at $x=0$. I have no problem with showing $f(x)$ is continuous at ...
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continuity and limits of $f(x,y)= \begin{cases} \frac{y\ln(x+1)}{y^2+(\ln(x+1))^2} &\text{if $y \neq 0$ }\\0&\text{if $ y=0$}\end{cases}$

Given the set $D:=\{(x,y) \in \mathbb{R}^2: x > -1\}$ and the function $f: D\rightarrow \mathbb{R}$ through $f(x,y)= \begin{cases} \frac{y\ln(x+1)}{y^2+(\ln(x+1))^2} &\text{if $y \neq 0$ ...
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36 views

The Inverse Function $f^{-1}(y)$

Let $$f(x)=\begin{cases} 1-x & \text{ if } -1<x\leq 0, \\ \frac{x^{-1}+\lfloor x^{-1}\rfloor}{1+x^{-1}+\lfloor x^{-1}\rfloor}& \text{ if } 0<x<1, \end{cases}$$ $\lfloor ...
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16 views

Continuity on two variable functions

I'm wondering if there's difference in saying a function with two variables being continuous for each variables and it's continuous for both variables? if so, what is the case for something like ...
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54 views

Is $ f(x,y) \mapsto \begin{cases} \frac{(x-y)^2}{(x-1)^2+(y-1)^2} \\0\end{cases}$ continuos?

$f: \mathbb{R}^2 \rightarrow \mathbb{R}, \begin {pmatrix} x \\ y \end{pmatrix} \mapsto \begin{cases} \frac{(x-y)^2}{(x-1)^2+(y-1)^2} &\text{if $(x,y) \neq (1,1)$ }\\0&\text{if $ ...
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42 views

How to define continuity of functions from $R$ to $P(R^2)$?

Consider a 2-dimensional amoeba that moves in $R^2$. This amoeba can be defined as a function $f$ from a real interval to $P(R^2)$: the real interval represents the time, and $P(R^2)$ (= the subsets ...
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23 views

Is the norm continuous? Can I switch limits like this?

I am in the middle of my proof and I want to know if the following is true, suppose $f_n$ is a Cauchy sequence, can i do this? If $$\| f_n(x) - f(x) \| \to 0,$$ then can I also say this limit is true ...