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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$f:X\rightarrow X$ be a continuous map, we need to show $f(\cap A_n)=\cap f(A_n)$

let $X$ be a complete metric space with metric $d$ and $A_{i}$'s are nested sequence of closed sets in $X$ i.e $[A_1\supseteq A_2\dots]$ such that $\sup\{d(x,y):x,y\in A_n\}\to0$ as $n\to\infty$ ...
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0answers
85 views

Prove that if $f$ is uniformly continuous then the one sided limit $\lim_{x\to 0^+} f(x)$ exists. [duplicate]

If $f(x)$ is a continuous function on $(0,1]$, prove that if $f$ is uniformly continuous, then the one sided limit $\lim_{x\to 0^+} f(x)$ exists.
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2answers
134 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
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3answers
106 views

Extension of a continuous map on $ {\mathbf{GL}_{n}}(\mathbb{R}) $ to $ {\mathbf{M}_{n}}(\mathbb{R}) $.

I was reading in my analysis textbook that the map $ f: {\mathbf{GL}_{n}}(\mathbb{R}) \to {\mathbf{GL}_{n}}(\mathbb{R}) $ defined by $ f(A) := A^{-1} $ is a continuous map. I also saw that $ ...
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0answers
71 views

A basic question on limit

Why for a continuous random variable $X$ there exists a $\delta > 0$ such that for all $x$ in $[c, c+\delta]$ the following is true: $$P(c < X \leq x) < \epsilon$$ for any given $\epsilon ...
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3answers
125 views

Question on continuity in Topology

$X=[0,2\pi)$ with the relative topology determined by the usual topology on $\mathbb R$, and the unit circle $S=\{(x_1,x_2)\mid x_1^2+x_2^2=1\}$, with the relative topology determined by the usual ...
5
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0answers
62 views

Functional sequence [duplicate]

Let $(f_n)$ be a sequence of functions $\mathbb{R} \rightarrow \mathbb{R}$. Suppose that for any $(x_n)$ convergent to $x$ we have $f_n(x_n) \rightarrow f(x)$. Prove that $f$ in continuous, there is ...
2
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1answer
78 views

equivalent metric

Let $(X; d)$ and $(Y; d')$ be metric spaces, and let $f : X \to Y$ be continuous. Define $df (x; y) = d(x; y) + d'(f(x); f(y))$ for $x, y \in X$. Show that $df$ is a metric on $X$ that is equivalent ...
4
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3answers
102 views

Consequence of Invariance of Domain

The Invariance of Domain theorem states that Given a continuous injection $f : U \to \mathbb{R}^n$, where $U$ is a nonempty open subset of $\mathbb{R}^n$, $f$ is an open map. These slides (see ...
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2answers
247 views

Continuously differentiable functions of bounded variation

From this question, we know that a continuous function of bounded variation is not necessarily absolutely continuous. But the example (Devil's staircase) given is not differentiable. What if we ...
2
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3answers
118 views

Example of a (dis)continuous function

The following thought came to my mind: Given we have a function $f$, and for arbitrary $\varepsilon>0$, $f(a+\varepsilon)= 100\,000$ while $f(a) = 1$. Why is or isn't this function continuous? I ...
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3answers
72 views

question about continuity

The question is: Assume $f$ is a bounded continuous function in $[a,b)$ and differentiable in $(a,b)$. Also assume $f$ is not continuous in $[a,b]$. prove: $f'(x)$ is neither upper bounded nor lower ...
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1answer
147 views

Question about L'hospital's rule and continuity

The question is: How can $f(0)$ be defined so the function $\displaystyle f(x)=(1+x^2)^{1/\large \tan(x)}$ will be continuous in $(-\pi/2,\pi/2)$? thanks I understood that i need to use l'hospital's ...
3
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1answer
149 views

Show that $\chi_A$ is continuous on $\operatorname{int}{A}$ and $A'$ but not $\partial_A = \overline{A} \cap \overline{A'}$

Let $A$ be a subset of $\mathbb{R}^n$. Show that the characteristic function $\chi_A$ is continuous on the interior of $A$ and of its complement $A'$, but is discontinuous on the boundary $\partial_A ...
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1answer
47 views

Continuous complex funtion

I have this function $$F(z)=\frac{1}{\alpha-i\sqrt{z}}$$ with $\alpha>0$ and the determination of the square root with $\Im z>0$. I have to study its continuity in the set $$A=\lbrace z|a\leq\Re ...
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1answer
47 views

Are the functions $f(x)g(x)$, $f(x)-g(x)$ and $\frac{f(x)}{g(x)}$ continuous when $x$ varies in a topological space?

Let $(X,\mathcal T)$ be a topological space and the functions: $$f:X\to \Bbb R$$ $$g:X\to \Bbb R^+$$ be continuous. Are the functions $f(x)g(x)$, $f(x)-g(x)$ and $\frac{f(x)}{g(x)}$ continuous?
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2answers
103 views

A question on a Lipschitz function

This is the problem: Prove or disprove the following statement: If $f:[0,+\infty]\rightarrow\mathbb{R^+}$ is a Lipschitz function and not bounded, then it has necessarily $\lim_{x\to+\infty} f(x) = ...
5
votes
2answers
113 views

Intermediate Value Theorem question

The temperature $T(x)$ at each point $x$ on the surface of Mars (a sphere) is a continuous function. Show that there is a point $x$ on the surface such that $T(x)=T(-x)$ (Hint: Represent the surface ...
2
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1answer
91 views

Continuous real-valued function and open subset

Let $f$ be a continuous real-valued function defined on an open subset $U$ of $\mathbb{R}^n$. Show that $\{(x,y):x\in{U},y>f(x)\}$ is an open subset of $\mathbb{R}^{n+1}$ Let ...
0
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1answer
94 views

Show that this function on $\mathbb{R}^2$ is not continuous at the origin (WITHOUT using limit test)

Consider a function defined on $\mathbb{R}^2$ by $$f(x,y) = \begin{cases} 0 & \quad \text{if $y\leq{0}$ or if $y\geq{x^2}$}\\ \sin\left(\frac{\pi y}{x^2}\right) & \quad \text{if ...
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3answers
404 views

Possible to have a continuous sequence?

I'm wondering if it's possible to have a continuous sequence $f: \mathbb{N} \to \mathbb{R}$? My intuition is telling me no because logically it would be impossible to map the natural numbers onto the ...
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38 views

Is Rayman function continuous and another question about the subject

The question as was written in title is whether the Rayman function is continuous where the Rayman function is this one: http://i.stack.imgur.com/xgFVy.png and another question (if I can): if I have ...
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3answers
148 views

$f: \mathbb{Q} \rightarrow \mathbb{R} \ \ \lim _{q \rightarrow t, \ q\in \mathbb{Q}} f(q) =g$

There's a continuous function $f: \mathbb{Q} \rightarrow \mathbb{R}$ Prove that $ \exists t \in \mathbb{R} \setminus \mathbb{Q} \ \ \exists g \in \mathbb{R} :\ \ \lim _{q \rightarrow t, \ q\in ...
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1answer
96 views

How can I study the continuty of this function?

Let $f\in L^2(\mathbb{R}^3)$ with compact support; is the function $$F(z)=\int_{\mathbb{R}^3}dx\bigg(f(x)\frac{e^{i\sqrt{z}|x|}}{4\pi|x|}\bigg)$$ continuous in the set $$Q=\lbrace{z: \Re z\in [a,b], ...
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2answers
86 views

Collecting definitions of continuity.

Let $X$ and $Y$ denote topological spaces and consider a function $f : X \rightarrow Y$. I'm collecting possible definitions/characterizations of the statement "$f$ is continuous." Here's two to get ...
2
votes
3answers
686 views

Uniform Continuity of $f(x)=x^3$

1.)Determine whether $f(x)=x^3$ is uniformly continuous on [0,2) So far, I have $\delta$ = 2 and $\epsilon$ = 8, and plan on using the sandwich theorem with $x^2$ and eventually equating $\delta = ...
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5answers
133 views

Let $\displaystyle f$ be a continuous function from $[0,4]$ to $[3,9].$

I came across the following problem that says: Let $\displaystyle f$ be a continuous function from $[0,4]$ to $[3,9].$ The which of the following options is correct? $1.$ there must be an $x$ ...
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1answer
243 views

Lipschitz and Holder continuous

Let a function $f(t,x)$, which is Lipshitz continuous in $x$ and $1/2$-Holder continuous in $t$. Is there any "official" and widely accepted way to denote the class of such functions ? I am ...
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2answers
248 views

Show that sawtooth function is continuous

Show that sawtooth function $f$ is continous, where $f$ is given by $f(n) = \left\{ \begin{array}{l l} x-2n & \quad \text{if } {2n \leq{x} \leq{2n+1}}, {n \in \mathbb{Z}}\\ 2n-x & ...
1
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1answer
113 views

Question on Continuous function and Lipschitz

$f:\mathbb{R}^n \to \mathbb{R}$ is contiuous. If $x \in \mathbb{R}^n$ and $C \in \mathbb{R}$ such that $f(x) < C$ ($C$ is constant). Prove that there is $r>0$ such that $\forall{y} \in B_r ...
10
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1answer
365 views

Additivity + Measurability $\implies$ Continuity

A function $f:\Bbb R \to \Bbb R$ is additive and Lebesgue measurable. Prove that $f$ is continuous. I know that on $\Bbb Q$, $f$ comes out to be linear. So, if $f$ is to be continuous then $f$ must ...
0
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1answer
59 views

$F(x,y)$ continues at $(x_0,y_0)$?

We have $F(x,y)=f(x), f(x)$ continues at $x_0$. How to prove that: for every $y_0 \in \mathbb{R}$, $F(x,y)$ continues at $(x_0, y_0)$?
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1answer
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Topology: continuous topological spaces

Let $X,Y,Z$ be topological spaces. Let $f:X\to Y$ be continuous, and let $g:Y\to Z$ be continuous. Prove $g\circ f:X\to Z$ is continuous. Use this fact to provide a detailed proof that homeomorphism ...
1
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1answer
61 views

Equintinuity of bounded linear functions equivalent to uniform boundedness

The claim is the following: Every family of bounded linear functions is equicontinuous if and only it is uniformly bounded. Equicontinuity is defined here. Any suggestions about this? I only ...
1
vote
1answer
88 views

Regarding Hölder continuity

Let $\alpha \geq 0$. We say that $f \colon D \to \mathbb{R}^m$ is $\alpha$-Hölder continuous if there is a constant $c$ such that for each $x,x_0\in D$, $|f(x) - f(x_0)| \leqslant c\cdot |x - ...
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2answers
279 views

Prove that $f$ is discontinuous at $(0,0)$

Let $f$ be defined by $$ f(x,y) = \begin{cases} \biggl\lvert \frac{y}{x^2} \biggr\rvert e^{-\bigl\lvert \frac{y}{x^2} \bigr\rvert} , \quad \text{ if $x \neq 0$} \\ 0, \qquad \qquad \quad \text{if $x ...
2
votes
2answers
264 views

Topology: Continuous Functions

Let $(X,T)$ be a topological space, and let $f,g\colon X\to \mathbb R$ be continuous, real-valued functions. Define the functions $f+g,f-g\colon X\to\mathbb R$ as follows: $(f\pm g)(x)= ...
2
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1answer
87 views

Prove $I:C([0,1])\rightarrow C([0,1])$ defined by $I(f)(x)=\int_0^x f(t)dt$ is uniformly continuous

Using $d_\infty(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, prove the above. I think I have a proof but it seems weird and I think it might be wrong. First, note that all functions in $C([0,1])$ are ...
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1answer
112 views

Prove that $f$ is continuous if and only if $f(\cdot+t) \to f$ pointwise as $t \to 0$

Let $f:\mathbb{R}\to \mathbb{C}$ be a function. Prove that $f$ is uniformly continuous if and only if $f(•+t) \to f$ in $L_{∞}$ as $t\to0$ Prove that $f$ is continuous if and only if $f(•+t) \to f$ ...
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1answer
64 views

If $f>0$ is discontinuous and bounded on $I$, is there a $g>0$ bounded such that $fg$ is continuous at least at one point?

Let $I\subseteq\mathbb{R}$ be a closed and bounded interval and let $$f:I\to(0,\infty)$$ be a bounded function that is discontinuous at every point in $I$. Does there exist a bounded function ...
4
votes
2answers
142 views

Showing if something is continuous in Topology

If $f : X \to \mathbb{R}$ is continuous, I want to show that $(cf)(x) = cf(x)$ is continuous, where $c$ is a constant. Attempt: If $f$ is continuous, then we want to show that the inverse image of ...
7
votes
4answers
201 views

Definition of continuity at a point

In "Principles of mathematical analysis" Walter Rudin gives the definition of $$\lim_{x \rightarrow p} f(x)=q$$ for $f: X \supset E \rightarrow Y$ with $X,Y$ metric spaces, in this way: for ...
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1answer
91 views

Suppose that $\displaystyle f(x)=\sin x,$ if $x \leq c$ and $\displaystyle f(x)=ax+b,$ if $x>c$

I came across the following problem which says: Let $a,b,c$ be non-zero real numbers. Also suppose that $\displaystyle f(x)=\sin x,$ if $x \leq c$ and $\displaystyle f(x)=ax+b,$ if ...
3
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5answers
458 views

I need to prove the continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof

I need to prove the continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof These is what I have so far but am not sure how to continue $|\log x-\log a| < \epsilon$ $\log a- \epsilon < ...
3
votes
4answers
118 views

What does continuity of inclusion means?

If $A,B,C$ are three spaces such that $A\subset B\subset C$ and $A$ is dense in $C$. Now my teacher said that the inclusion between three spaces are continuous and so you can directly say that $B$ is ...
3
votes
1answer
261 views

Piecewise function $f(x,y)$ - Limits and Continuity

Consider the function $f : \Bbb R^2 \to \Bbb R$ given by: $$f(x, y) = \begin{cases} 0 &\text{if } x < 0,\\ x\cdot y &\text{if } x \geq 0 \text{ and }y \geq 0,\\-x &\text{if } x \geq 0 ...
1
vote
1answer
78 views

Continuity and Integrability

Any help with this problem is appreciated. Consider the function $f(x) = \sum_{n=1}^\infty x n^{-\beta} e^{-nx}$. For what values of $\beta \in \mathbb{R}$ is $f$ continuous on $[0,\infty)$ and in ...
2
votes
1answer
98 views

Proving something is continuous in Topology

If $f,g : X \to \mathbb{R}$ are continuous, prove the following function is also continuous from $X$ to $\mathbb{R}$. $f + g$ defined by $(f + g)(x) = f(x) + g(x)$. This is for topology, so my first ...
2
votes
0answers
25 views

Monotonic function non-continuous in each rational [duplicate]

How can I prove that exists a monotonic non-decreasing function $f: [0,1] \rightarrow \mathbb R$ that isn't continuous in every rational of its domain?
5
votes
2answers
267 views

Uniformly continuous function acts almost like an Lipschitz function?

Can anyone help? Let $g:I \to \mathbb{R}$ be an uniformly continuous function, where $I$ is an interval. Prove that exists an constant $c$ that satisfies: $$\lvert g(x)-g(y)\rvert < 1 + c \lvert ...