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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continuity of linear functional on family of functions

If $A$ : $C[a,b]\rightarrow \mathbb{R}$ is a continuous linear functional, then $ t\mapsto A(f_{t})$ is a continuous function on $\mathbb{R}$. where \begin{align} f_t(x)= \left\{ \begin{array}{lr} ...
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If $f_n \to f$ uniformly and $f$ is continuous, does that imply $f_n$ is continuous?

I have a theorem in my book which says if $(f_n)$ is a sequence of functions uniformly converging on $A$ to $f$, and is continuous at some point $c \in A$, then $f$ is also continuous at this point ...
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Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
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1answer
41 views

Continuity of the locus of the maximum of a two variables real function

Suppose that $$\begin{array}{lrcl} f : & [0,1]^2 & \longrightarrow & \mathbb{R} \\ & (x,y) & \longmapsto & f(x,y) \end{array}$$ is a continuous function and that for all $x ...
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piecewise defined function finding at which points it is continuous

We have: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ $$ f(x) = \begin{cases} x^3 - 3x + 2 &\text{if }x \in \mathbb{Q} \\ x^3 + x^2 + 4 & \text{if }x \in\mathbb R\setminus\mathbb Q \\ ...
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Functional limits and definition of continuity - difference and implications?

Continuity: A function $f : A → \mathbb{R}$ is continuous at a point $c ∈ A$ if, for all $\epsilon > 0$, there exists a $δ > 0$ such that whenever $|x − c| < δ$ (and $x ∈ A$) it follows that ...
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Does $\Vert f-s_n \Vert_\infty \to 0$ still hold for $f\in C^0[a,b]$?

If $f\in C^2[a,b]$ and $s_n$ its piecewise linear interpolation at points $x_0, \ldots, x_n$ with $h_n = \max_{j=0,\ldots,n-1} (x_{j+1}-x_j)$ then one can show that $$\Vert f-s_n \Vert_\infty \leq ...
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$T$ continuous in $x_0$ then $T$ is continuous

Let $T:V\to W$ be a linear operator, with $V, W$ normed spaces. Show that if there exist $x_0 \in V$ such that $T$ is continuous in $x_0$ then $T$ is continuous. I'm thinking that given an ...
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49 views

Are bounded analytic functions on the unit disk continuous on the unit circle?

Let $f(z)$ be holomorphic on the open disk $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Moreover, let $f$ be bounded on the boundary of $\mathbb{D}$, i.e. $$ \sup_{\varphi \in [0,2\pi]} ...
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Prove that the difference of continuous and monotonically increasing functions has continuous variation

Let $G:[0,\infty)\to\mathbb{R}$ be continuous and $$V^1_t(G):=\sup\bigcup_{n\in\mathbb{N}}\left\{\sum_{i=0}^{n-1}\left|G_{t_{i+1}}-G_{t_i}\right|:0=t_0\le\cdots\le t_n=t\right\}$$ be the variation ...
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1answer
28 views

Prove that a function is continuous using basic open sets

Using basic open sets of $\Bbb R$, prove that $f(x,y,z)=x^2+y^2+z^2+2x+2y+6$ is a continuous function from $\Bbb R^3$ to $\Bbb R$. My attempt: Since $f(x,y,z)$ is continuous and $f(x,y,z)\in ...
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55 views

Does there exist a model of $ZF¬C$ in which there is a function on $\mathbb R$ which is sequentially continuous at a point where it is not continuous? [duplicate]

Does there exist a model of $ZF¬C$ in which there is a function $f:\mathbb R \to \mathbb R$ such that $f$ is sequentially continuous at some $a \in \mathbb R$ but not $\epsilon-\delta$ continuous , ...
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43 views

“Sequential continuity is equivalent to $\epsilon$-$\delta $ continuity ” implies Axiom of countable choice for collection of subsets of $\mathbb R$?

"A function $f: \mathbb R \to \mathbb R$ is continuous at $x \in \mathbb R$ , if and only if it is sequentially continuous " , does this statement imply "the Axiom of Choice for countable collections ...
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2answers
110 views

Can the Intermediate Value Theorem be proved by Heine-Borel Lemma?

Can the Intermediate Value Theorem be proved by Heine-Borel Lemma, and how? I mean "Every open cover of close interval has a finite subcover", without compactness etc. Because in class we proved it ...
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1answer
49 views

If $f:\mathbb{R}\rightarrow\mathbb{R}$, $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0$ then $\frac{f(x)}{x}\in C^{\infty}(\mathbb{R})$

If $f:\mathbb{R}\rightarrow\mathbb{R}$, $f\in C^{\infty}(\mathbb{R})$ and $f(0)=0$ then $\frac{f(x)}{x}\in C^{\infty}(\mathbb{R})$. Following ther is what i did: the Maclaurin series for $f(x)$ is ...
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3answers
71 views

For any closed set $A$ of $\mathbb R$ , does there exists a function $f:\mathbb R \to \mathbb R$ such that, $f$ is discontinuous exactly on $A$?

For any closed set $A$ of $\mathbb R$ , does there exists a function $f:\mathbb R \to \mathbb R$ such that, $f$ is discontinuous at every point in $A$ but, is continuous at all other points ?
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1answer
25 views

Why is this image sequentially compact? [duplicate]

Assuming $X$ and $Y$ are normed spaces, $K\subset X$ and $f:K\rightarrow Y$. Why is the image $f(K):=\{f(x)\in Y: x\in K\}$ sequentially compact, if $K$ is sequentially compact and $f:K\rightarrow Y$ ...
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2answers
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$f : \mathbb R \to \mathbb R $ be a differentiable and $f'(x) $ is bounded function then $f$ is unbounded.

$f : \mathbb R \to \mathbb R $ be a differentiable function with $f(0) =0$. If for all $x \in \mathbb R , 1< f'(x) < 2$ then $f$ is unbounded. We know that when $f'(x) > 0$ then $f$ is ...
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Absolute continuity equivalence

Show that a function $f:[a,b] \rightarrow \mathbb{R}$ is absolutely continuous if and only if there exists a sequence of Lipschitz functions $(f_n)$ such that $V([a,b], f-f_n)$ converges to $0$.
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Differences between a quotient map and a continuous function in topology

Def. for a continuous function: Let $X$ and $Y$ be topological spaces. A function $f : X \rightarrow Y$ is continuous if $f^{-1} (Y)$ is open in $X$ for every open set $V$ in $Y$. Def. for ...
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25 views

Express continuity on multiple intervals using union?

Can I express the continuity on multiple intervals using union? For example, I want to discuss the continuity of $f(x)=\frac{1}{x}$. Can I say that it's continuous on $(-\infty,0)\cup(0,+\infty)$ ...
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upper and lower semi continuouis

let $X$ be a topological space,$ f_n:\mathbf X \to \mathbb R$ is sequence of lower semi continuous functions then the $ \sup \{f_n\}=f $ is also lower semi continuous proof: f is ...
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1answer
24 views

Continuity of partial derivatives only along their axis?

My main question for which I will give an example right below is whether for a partial derivative to exist at a point (say $\frac{\partial f}{\partial x}$) it is necessary for it to be continuous at ...
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2answers
41 views

Clarifications about the correct way to solve exercises (continuity, partial derivatives, differentiability)

I need some clarifications about the correct way to solve an exercise. I have this function: $$f(x,y)=\frac{(x-1)y^2}{\sin^2\sqrt{(x-1)^2+y^2}}$$ and I have to analyse the existence of partial ...
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Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability [duplicate]

Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability $f:\mathbb{R}^n\rightarrow\mathbb{R}$ I don't know how to star this problem, I just know the definition ...
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Is QR decomposition of an invertible matrix “continuous”? [closed]

I know the QR decomposition is unique for invertible matrix. So, there is a well-defined map $$A \to (Q,R)$$ where $A \in GL^+(n, \mathbb R)$ , $Q\in O(n)$ and $R$ is an upper triangular matrix with ...
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can any one prove the second branch of the function

can any body tell me how the 2nd branch of the function is defined here $f_t(x)$ is a continuous function \begin{align} f_t(x)= \left\{ \begin{array}{lr} \dfrac{x^t}{t(t-1)\ldots(t-n+1)} & ...
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1answer
46 views

$f \colon [a,b] \to [a,b]$ continuous has exactly one fixed point if $f'(x) \neq 1$ for all $x \in (a,b)$

Let $f$ be a continuous function from [a, b] to [a, b], and is differentiable on (a, b). We will say that point y $\in$ [a, b] is a fixed point of f if $y = f(y)$. If the derivative $f'(x) \neq 1$ ...
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3answers
41 views

continuity of |f| and its f [duplicate]

I want to know wether if |f| is continuous at every point of R, then f is continuous also or not? and what about its converse? if f is continuous in every point of R, does that means |f| also ...
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4answers
29 views

Lipschitz Continuous functions

I am currently doing some revision for an exam next week when I came across this question from 3 years previous. I am a bit confused on how to tackle this question and lack a thorough understanding of ...
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1answer
32 views

Continuous random variable pdf question.

The continuous random variable X has pdf where $$f(x) = \begin{cases} \frac{25}{12(x+1)^2},\quad & 0\le x\le 4 \\ 0 & \text{otherwise} \end{cases} $$ $E(X+1) = 1\frac23$ and $E(X) = 2/3$ ...
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Proof coutinuity

$f:=\left\{\begin{matrix} \frac{xy}{\sqrt{x}+y^2}, x,y\neq 0\ begin & \\ 0, x,y=0 & \end{matrix}\right.$ Is f continuous in (0,0)? My idea is: $\left | f(x,y) \right | = \left | ...
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1answer
99 views

Alternate proof for Arzela-Ascoli

Im trying to finish a beautiful excercise, which consist of giving an alternate proof for the following corollary of Arzela-Ascoli´s Theorem. Given $X,Y$ metric spaces, $X$ compact, $Y$ complete, and ...
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Let $I$ be a line interval and $f: I \to \mathbb{R} $ continuous show that if $f(I) \subseteq \mathbb{Q} $ then $f$ is constant.

Since $f$ is continuous it maps intervals to intervals. Since $f(I) \subseteq \mathbb{Q} $ no element of $\mathbb{R} \setminus \mathbb{Q}$ is element of $f(I)$. The irrationals are dense in ...
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33 views

Periodic function $f: \mathbb R \to \mathbb R$ (continuous if possible ) such that the sequence $\{f(n) \}$ is not constant and convergent?

Does there exist a periodic function $f: \mathbb R \to \mathbb R$ such that the sequence $\{f(n) \}$ is not constant and convergent ? Does there exist such a continuous function ?
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1answer
38 views

Preimage of a closed set is a closed implies f is continuous. Some concerns about the proof

Ok. I have managed thru: If f is continuous then the preimage of open set is a open set If the preimage of the open set is a open set then f is continuous If f is continuous then the preimage of a ...
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111 views

Continuity of normalized displacement vector for a smooth closed curve

I am currently working on chapter 3.12 of "Differential Equations and Dynamical Systems" by Lawrence Perko. I am stuck on the continuity of the function $g$ in Theorem 3. My work (up to the prove of ...
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Characterization of continuity in terms of filters

Two characterizations of continuity are For all filters $\mathcal{F}$, $\mathcal{F} \to x \implies f(\mathcal{F}) \to f(x)$. $f(\overline{S}) \subseteq \overline{f(S)}$ for all $S$ ...
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38 views

For what values of $t$ is $f(x)$ differentiable at $x = 0$? [closed]

Given $$f(x) = \begin{cases} x^t \sin \frac{1}{x^2} & \text{if } x\ne0 \\ 0 &\text{if } x=0 \ \end{cases}. $$ For what values of $t$ ($t$ is real) $f(x)$ is differentiable at $x = 0$? I ...
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Conceptual question regarding convergence and continuity

Let $X,Y$ be metric spaces. Let $(f_n)$ be a sequence of functions from $X$ to $Y$ equicontinuous that converges pointwise to a function $f:X \to Y$. Then, $\{f,f_1,f_2,...\}$ is equicontinuous. ...
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Have you ever seen this result about pointwise/uniform convergence of a net of continuous functions?

I am in need of results transforming pointwise convergence of functions into uniform convergence. Since I wasn't satisfied with Dini's theorems, I had to prove the following result: Let $K$ be a ...
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How do you call a metric space with “continuous” points?

I have the impression that "continuous space" is not a mathematically precise concept (as opposed to continuous functions that can be defined under various contexts). However, I find I need to refer ...
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46 views

Connection between $\epsilon-\delta$ definition of limit and Weierstrass definition of continuous functions

I have just read the definition of a continous function according to Weierstrass, and for me it seems the same definition of the $\epsilon-\delta$ definition for limits. I also know that there exists ...
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3answers
67 views

Continuity of a function $f: \mathbb{R}^2 \to \mathbb{R}$

It's easy to check that the function $$ f_1(x, y) = \begin{cases}\frac{x y}{x^2 + y^2} &\text{if (x, y) ≠ (0, 0)}\\0&\text{if (x, y) = (0, 0)}\end{cases}$$ is not continuous in $0$, because ...
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1answer
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Implicit function theorem application: $h(f(v),v)=0$ find $f(v)$…

I am preparing for an exam and I cannot seem to figure out how to solve exercises where I need to apply the implicit function theorem. Exercise: Let $h: \mathbb{R}^2 \rightarrow \mathbb{R}$ given ...
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23 views

Question about Rudin's Functional Analysis Closed Graph Theorem

In page 51 of Rudin's Functional Analysis, the closed graph theorem is proven, which says that if you have a linear map between two F-spaces whose graph is closed in the product space, then the map is ...
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31 views

Completion of C(I) to $L^{2}(I)$ for some arbitrary interval I

As $L^{2}(I)$ is the completion of C(I), without too many issues (as $L^{2}(I)$ is a space of equivalence classes with equivalence relation defined as functions equivalent if differ at only finitely ...
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69 views

Is there a name of such functions?

Let $U$ be an open subset of $ \mathbb R^n$ and consider $f :\mathbb R^n \to \mathbb R$ with the properties that $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions: Is there ...
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1answer
34 views

What implies $f_n (y) \leq f(x) + \epsilon$ about $f$ ?

Let $X$ be a regular topological space. Question: For which functions $f : X \rightarrow \mathbb R$, can we find a sequence of functions $f_n : X \rightarrow \mathbb R$ such that: $\forall ...
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2answers
52 views

What is the definition of differentiability?

Some places define it as: If the Left hand derivative and the Right hand derivative at a point are equal then the funtion is said to be differentiable at that point. Others define it based on ...