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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Convergence in Sobolev Spaces

Consider the bounded mapping $A:W^{1,p}(\Omega) \rightarrow W^{1,p}(\Omega)^{*}$ where $A$ is defined as: $\langle A(u),v \rangle\text{ } := \int_{\Omega}a(x,u,\nabla u)\cdot \nabla v + c(x,u,\nabla ...
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1answer
84 views

Verifying continuity of the deformation retraction of the mapping cylinder

Given $f : X \rightarrow Y$ I have the mapping cylinder $M_f = (X \times I) \sqcup Y / \sim$ where $I = [0,1]$ and $(x,0)_1$ is glued to $f(x)_2$ (I'd like to be pedantic for this verification so I'm ...
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67 views

Equivalent Definition of Continuity

I am trying to understand a proof in a book. (Willard, General Topology, p. 45) Theorem: Let $X$ and $Y$ be topological spaces and $f : X \to Y$. If for each $E \subseteq X$, ...
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1answer
43 views

Functions that preserve continuity (of functions with a fixed domain) via composition

Let $X$, $A$, and $B$ be topological spaces. I'm interested in functions $f:A\rightarrow B$ that preserve continuous functions with domain $X$; that is functions $f$ such that $f\circ ...
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38 views

Prove that a Particular Set Is an Interval

Suppose that $I$ is an interval in $\mathbb R$, and $f: I \rightarrow \mathbb R$ is continuous on I. My text (Stoll, Introduction to Real Analysis, 2nd Ed) proved that $f(I)$ is an interval by ...
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29 views

$f(x)=\int_{0}^{x} \{5+|1-y|\}dy$ if $x<2$

$f(x)=\int_{0}^{x} \{5+|1-y|\}dy$ if $x<2$ $f(x)=5x+2 $ if $x\ge 2$ I need to check continuity and differentiability at $x=2$ $\lim_{x\uparrow 2}f(x)=\int_{0}^{2} ...
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Prove T is continuous on $C^{(1)}$

let $x(t)$ function defined on $C_{[a,b]}$ defined on [$a,b$] $T_{1}(x)$ = $\phi$[$x(t_{0}),x(t_{1}),...,x(t_{n})$] and $\phi$ is continuous. $T_{2}(x)$ = $\int_{0}^{1}$ $\sqrt{1+ (x'(t))^{2}}$ $dt$ ...
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3answers
128 views

Do continuous mappings always have an inverse?

There's a theorem that states that a mapping $f$ from $X$ to $Y$ is continuous if and only if the inverse image of any open set in $Y$ is open in $X$. Does this mean that continuous functions always ...
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104 views

Is every $G_\delta$ set the set of continuity points of some function $f$?

I can prove that given a function $f:X \rightarrow Y$, where $X,Y$ are metric spaces, the set $A \subseteq X$ of points on which $f$ is continuous, is $G_{\delta}$. (Take $U_n = \bigcup_{y \in ...
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Proving that a certain continuous function is surjective.

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $|f(x)-f(y)|≥|x-y| ,\forall x,y \in \mathbb R $ , then how do we prove that $f$ is surjective ?
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Let $f$ be with $\lim_{|x|\to\infty} f(x) = 0$, show that: $\exists x_0\in\mathbb{R}: \, \mid f(x)\mid \, ≤ \, \mid f(x_0)\mid$.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous such that $$\lim_{x\to\infty} f(x) = \lim_{x\to -\infty}f(x) = 0$$ Show that there $\exists x_0\in\mathbb{R}: \, \mid f(x)\mid \, ≤ \, ...
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0answers
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Sard's theorem and the measure of the set of critical points

The origin of my question is derived from the following theorem Theorem: Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a map of class $\mathcal{C}^1$. Then, the following phrases are equivalents [1]: a) ...
3
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1answer
114 views

Weak continuity in Sobolev Spaces

First consider the following two Sobolev Embedding Theorems. Theorem 1: The continuous embedding $W^{1,p}(\Omega) \subset L^{p^{*}}(\Omega)$ holds provided the exponent $p^{*}$ is defined as ...
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1answer
37 views

Show that $f (a,b) \rightarrow \mathbb{R}$ is uniform continous if and only if there is a continous $g|_{(a,b)} = f$.

The assignment is: Let $a,b \in \mathbb{R}$ with $a < b$ and $f: (a,b) \rightarrow \mathbb{R}$. Show that: The function $f$ is uniform continous if and only if there exists a continous function ...
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A detail in theorem 6.9 rudin real analysis

here is the theorem(Thm 6.9 page 126): if $f$ is monotonic on $[a,b]$ and if $\alpha$ is continuous then $f \in \mathscr R$ ($\alpha$ is assumed montonic) the detail that i don't understand in the ...
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0answers
35 views

Lagrange MVT and finding function [duplicate]

I have two questions (use of IVT,MVT and derivatives is allowed, integrals are not allowed as well as Riemann integration): First question: Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable ...
3
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1answer
43 views

Non-polynomial $C^{\infty}$ function $f:\mathbb{R}\rightarrow\mathbb{R}$ with rational values for rational arguments?

Let's say that $ \ f : \mathbb{R} \rightarrow \mathbb{R} \ $ and $ \ f \ $ is $C^{\infty}$ function. Assume that for every $ \ x \in \mathbb{Q} \ $ we have also $ \ f(x) \in \mathbb{Q}$. Are the ...
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Continuity of $f(x)= x\cos 1/x$ when $x\neq 0$, and $0$ otherwise

Is the following function continous on $\mathbb R$? $f(x)=\begin{cases}\begin{align} &x\cos \frac 1x, & x\neq 0 \\ &0, & x=0 \end{align}\end{cases}$ I tried to derive it and show ...
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1answer
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Is delta distribution continuous and differentiable with dual space norm?

I know that delta distribution $\delta : \mathcal S (\mathbf R) \to \mathbf C$ is continuous with usual seminorm and here. I am interested in its continuity with dual-space $H^{-1}(\Omega)$ of ...
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Why is this Takagi's function continuous?

1903 Takagi constructed the function $f: [0,1] \rightarrow \mathbb{R}$ with $f(x) := \sum_{k=0}^\infty 2^{-k} \mathrm{dist}(2^k x, \mathbb{Z})$ where $\mathrm{dist}(x,A) := \inf\{|x-y| : y \in A\}$ ...
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55 views

minimum/maximum of two limits

Let $f: \mathbb {R} \rightarrow \mathbb {R} $ continuous so that $v =\lim\limits_{x \rightarrow -\infty} f(x) $ and $w = \lim\limits_{x \rightarrow \infty} f(x)$ are existing. I want to show that ...
3
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2answers
58 views

Prove uniform contininuity (probably by Lipschitz continuity)

Prove uniform continuity at $(0,\infty)$ for: $$f(x) = x + \frac{\sin (x)}{x}$$ Derivative is: $$f'(x) = \frac{x\cos (x) - \sin (x) + x^2}{x^2}$$ so, taking the limit at $\infty$ I got the value of ...
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1answer
73 views

Continuity of $h(x)=f(x) \cdot g(x)$

$h(x)=f(x) \cdot g(x)$ I want to check whether this function is continuous in its domain $\mathbb{R}$ or not. definition by cases: $f(x)$ and $g(x)$ are both continuous $\Rightarrow f(x) \cdot ...
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3answers
111 views

Continuity of $f(x)=\max\{x,0\}$

$$f(x)=\max\{x,0\}$$ I want to check whether this function is continuous in its domain $\mathbb{R}$ or not, but unfortunately I have no idea how to start.
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2answers
122 views

Example of continuous function that isn't uniformly continuous and isn't 1/x

I understand that in an open interval the only functions that are continuous but not uniformly are functions whose limits are singularities. But when we have a function $f:H\rightarrow\mathbb{R}$ and ...
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Proving and disproving $\exists b,c\in \mathbb R$ such that $f(x)=\frac a2x^2+bx+c $

Let $f:I\to\mathbb R$ where $I$ is an interval, $f''(x)=a \ \ \forall x\in I$. Prove that there exsits such numbers $b,c\in \mathbb R$ such that: $f(x)=\frac a2x^2+bx+c ,\ \forall x\in I$. ...
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1answer
42 views

Proving this Taylor-esque expansion for a $C^2$ function vanishing at 0 and 1

I am trying to prove the following (which I think is true!): if $f:[0,1]\rightarrow \mathbb{R}$ is twice continuously differentiable and $f(0)=0=f(1)$, then for every $x \in (0,1)$ there exists $\xi ...
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2answers
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Continuity of the function

Is the sequence a continuous function on the set of natural numbers? My book on complex numbers insists that for the function to be continuous, the limit at a point must exist, which, of course, makes ...
2
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1answer
128 views

$x_n$ convergence to $x$ implies $f_n(x_n)$ convergence to $f(x)$. prove that $f$ is continuous

Let $f$ and $f_n$ be functions from $\mathbb{R} \rightarrow \mathbb{R}$ Assume that $f_n (x_n) \rightarrow f (x)$ as $n\rightarrow \infty$ whenever $x_n \rightarrow x$. Prove that $f$ is ...
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Prove $x^{2 \over 3} \ln(x)$ is uniformly continious

Prove $x^{2 \over 3} \ln(x)$ is uniformly continuous in $(1,\infty)$ To my understanding I need to show the derivative is bounded. That will prove uniform continuity. The derivative is: $$ ...
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3answers
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Compact subsets of a metric space

I am trying to to prove that f: X --> Y is continuous on X if and only if f is continuous on every compact subset of X. X and Y are metric spaces. How do I show that every point of X belongs to some ...
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94 views

Redefine Function to Solve Discontinuity

$$ f(x) = \frac{6x^2-5x-4}{2x^2+x} $$ $f(x)$ is discontinuous at $x = -1/2$ Redefine $f(-1/2)$ so that the discontinuity can be removed
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1answer
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Prove $f(x) = \sqrt {\ln x} \ln (\ln x)$ is uniformly continuous.

Let $f(x) = \sqrt {\ln x} \ln (\ln x)$ Prove $f(x)$ is uniformly continuous. I'd be glad to get hint/guidance. I tried to follow the definition of uniformly continuous, but got stuck in the very ...
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1answer
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Relationship between f(X) and f(closure of X)

I am trying to prove if f is continuous and closed ("closed" means the image of any closed subset of the domain is closed) then f(closure of X) equals the closure of f(X). I was able to prove that if ...
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1answer
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Basic Continuity issue.

F is differentiable . We know that : Lim F(x^2) = F(0) when x^2 -> 0 How can we show in an easy way that : Lim F(x^2) = F(0) when x->0 Can we derive this directly from continuity, no ...
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1answer
53 views

is continuity preserved under Expectation?

Let's say I have a random function $X(t)$ that is continuous in $t$, almost surely. Is it true that $$\mathbb E(X(t_1)) = \mathbb E\left(\lim_{t\to t_1} X(t)\right)?$$ This seems incorrect to me ...
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1answer
74 views

Difference between expressions regarding Lipschitz continuity

Let $f:\mathcal{I}\times \mathcal{X} \to\mathbb{R}$ be an arbitrary function, e.g., $f(t,x)=t^2+x$. What are the differences between the following locally Lipschitz continuity definitions: "$f$ is ...
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Difference between expressions regarding continuity

Let $f:\mathbb{R_{\geq 0}}\times \mathbb{R}^n\to\mathbb{R^n}$ be an arbitrary function, e.g., with $n=1$, $f(t,x) = t^2+x.$ What is the difference among the following expressions: "$f$ is continuous ...
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How do the Fourier Transform of sampling and the Frequency-domain convolution match?

The Fourier Transform(FT) is $X(\upsilon) = \int_{-\infty}^{\infty}x(t)e^{-2{\pi}i{\upsilon}t}dt$. The impulse train is $\delta_1(x)=\sum\limits_{k=-\infty}^{\infty}\delta(x-k)$, and its FT is ...
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Continues function so lim(x)=lim(x^2) x->0?

Let $F$ be a continuous function on $\mathbb{R}$. Can we derive from here that $\lim_{x \to 0} F(x)= \lim_{x \to 0} F(x^2)$? I think that it's true because of Heine But I can't find a way to prove it ...
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2answers
59 views

Convergence of $f(p_n)$ insufficient to show continuity?

I came across a problem that asked one to assume $f: M\to N$ is a function from a metric space to another, and that if $(p_n)$ in $M$ converges then $f(p_n)$ in $N$ converges. It asked that one show ...
4
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1answer
106 views

Proving continuity of exp(x)

Well, my teacher went through a method of proving continuity of $\exp(x)$ which I don't like, so I tried to go about it a different way: We have proved the following (which I use) $\exp(x+y) = ...
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Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
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Continuity in multivariable calculus

I want to find out the points, where the function $f(x,y)=\dfrac{xy}{x-y}$ if $x\neq y$ and $f(x,y)=0$ otherwise, is continuous. I have shown that at all the points $(x,y)$, where $x\neq y$, $f$ is ...
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(…) Show that $f(x) ≤ f(b)$ for all $x \in [a,b]$ and that $f(a) = f(b)$

Can someone help me with this proof? Let $a,b \in\mathbb{R}$ with $a < b$. Furthermore let $f: [a,b] \rightarrow \mathbb{R}$ be continuous with the following conditions: For $\forall x ...
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1answer
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Constructing explicit lift of a circle homeomorphism

Studying a book by Luis Barreira in the Universitext Collection -- Dynamical Systems: an Introduction -- I'm told that given $f: S^{1} \to S^{1}$ homeomorphism, it's always possible to construct a ...
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235 views

Showing that the function is continuous but not differentiable

Let $$ f(x,y) = \begin{cases} \dfrac{xy}{\sqrt{x^2+y^2}} & \text{if $(x,y)\neq(0,0)$ } \\[2ex] 0 & \text{if $(x,y)=(0,0)$ } \\ \end{cases} $$ Show that this function is continuous but not ...
2
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3answers
199 views

Is $\log (1 + {x^2})$ uniformly continuous on $[0,\infty)$? [duplicate]

Is $\log (1 + {x^2})$ uniformly continuous? Here is my attempt: Let $\forall\left| {x - y} \right| < \delta$: $\left| {\log (1 + {x^2}) - \log (1 + {y^2})} \right| = \left| {\log (\frac{{1 + ...
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2answers
84 views

Prove/Disprove $f(x) = x + \frac{x}{{x + 1}}$ is uniformly continuous at $\forall x,y \in [0,\infty )$ [duplicate]

Prove/Disprove $f(x) = x + \frac{x}{{x + 1}}$ is uniformly continuous at $\forall x,y \in [0,\infty )$ This is my trial: $$\forall \varepsilon > 0\exists \delta > 0.\forall x,y \in ...
2
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3answers
182 views

Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$

Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$ Going from the epsilon delta definition we get: $$\forall x,y>1,\text{WLOG}:x>y \ ,\ \forall\epsilon>0,\exists\delta>0 ...