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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1answer
24 views

Partial derivatives and differentiability, continuity

Function $f : \mathbb{R}^3 \rightarrow \mathbb{R}$ has in every $x$ of domain partial derivatives $\frac{\partial f}{\partial x_1}(x) =x_2$, $\frac{\partial f}{\partial x_2}(x) =x_1$, $\frac{\partial ...
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0answers
35 views

Continuous function with support continuously embedded [duplicate]

Can someone give me a solution for this? We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ ...
1
vote
1answer
59 views

Is the complement of the closed unit disk in the plane homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ? [closed]

Is $\mathbb R^2 \setminus D^2$ , where $D^2=B[0;1]$ is the closed unit disk , homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ?
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5answers
700 views

Is the plane minus a line segment homeomorphic with punctured plane?

Is $\mathbb R^2$ minus a line segment i.e. $\mathbb R^2 \setminus ([0,1]\times \{0\}) $ homeomorphic with a punctured plane $\mathbb R^2\setminus \{(0,0)\}$ ?
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1answer
38 views

Let $f$ be a continuous and positive function on $\mathbb{R}_{+} $ such that $\lim_{x \to \infty} <1$

Let $f$ be a continuous and positive function on $\mathbb{R}_{+}$ such that $\displaystyle\underset{x \to \infty}{\lim} \frac{f(x)}{x} <1$. Prove the equation $$f(x)=x$$ has at least one solution ...
0
votes
1answer
84 views

Example of a jump discontinuity where the left and right hand limits do not exist? [closed]

Right off the bat I should probably mention that I am speaking more visually rather than in manners that can be proven rigorously. Please keep that in mind when reading. I'm looking for a function ...
8
votes
3answers
650 views

Why is/isn't the derivative of a differentiable function continuous?

I am confused about the following Theorem: Let $f: I \to \mathbb{R}^n$, $a \in I$. Then the function $f$ is differentiable in $a$ if and only if there exists a function $\varphi: I \to \mathbb{R}^n$ ...
2
votes
1answer
41 views

On the matter ; If $f:X \to Y$ is a function with closed graph and compactness preserving then $f$ is continuous

Let $X,Y$ be metric spaces , $f:X \to Y$ be a function , with closed graph , carrying compact sets to compact sets ; then I claim that $f$ is continuous Proof: Let , if possible , $f$ be not ...
0
votes
1answer
60 views

Help proving or disproving the following

Let $X,Y$ be topological spaces. Suppose $X=\bigcup_{\alpha\in\Lambda}A_\alpha$ for $\{A_\alpha\}_{\alpha\in\Lambda}$ closed in $X$, then Find a function $f:X\to Y$ such that for all $\alpha\in\...
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0answers
19 views

$f$ is of Baire class $\xi$ implies existence of a topology such that $f$ is continuous with respect to that topology

Suppose that $(X,\tau)$ are $Y$ are Polish spaces and a function $f:X\rightarrow Y$. Show that $f$ is of Baire class $\xi$ if and only if there is a Polish topology $\tau^{\prime} \supset \tau$ with $\...
2
votes
2answers
42 views

Interior of a preimage of a continuous function

Let $ f:\mathbb{R}^n\rightarrow \mathbb{R} $ be convex. Let there exist a point $ x_0 $ with $ f(x_0)<0 $. Prove that $$ \operatorname{int}\left\lbrace f(x)\ge 0 \right\rbrace = \left\...
1
vote
1answer
15 views

Lipschitz-continuity of a particular function

I have the following question. Let $ g_1,\ldots,g_k: \mathbb{R}^n\rightarrow \mathbb{R} $ be Lipschitz continuous (with respective constants $ L_1,\ldots,L_k>0 $). How can I proove the Lipschitz-...
2
votes
1answer
36 views

Examine whether the function $f(x, y)$ is continuous on $\Bbb R^2$ or not

Given, $f: \Bbb R^2 \rightarrow \Bbb R,$ $$f(x, y) := |\frac y {x^2}| e^{-|\frac y {x^2}|}, x \neq 0, y \in \Bbb R,$$ $$f(x, y) := 0, x = 0,$$ I have to decide whether the function ...
1
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2answers
77 views

$f$ be a function on real line carrying compact sets to compact sets and fiber of every point under $f$ is closed , is $f$ continuous ?

Let $f:\mathbb R \to \mathbb R$ be a function such that it carries compact sets to compact sets and $f^{-1}(\{x\})$ is closed for every $x \in \mathbb R$ , then is $f$ continuous ? (I know that if $...
0
votes
2answers
57 views

Let $f$ a continuous function on $[a,b]$ such $f(a)=f(b)$ [closed]

Let $f$ a continuous function on $[a,b]$ such $f(a)=f(b)$ Prove that: the equation $$f(x) = f\left(x+\frac{b-a}{2}\right)$$ has at least a solution in $[a,b]$ I tried to use Tvi(intermediate value ...
0
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1answer
45 views

(Conceptual) Continuity of binary relation $\succsim$ and definition using contour sets

Some background information: $\succsim$ is a binary relation that represents preference between two goods. $\succsim$ means "x is at least as good as y." Continuity of this relation is defined to be ...
1
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2answers
63 views

Is the empty set a topological space? [duplicate]

If so, is the empty function from it to any other space considered a continuous function? I can't really convince myself either way.
0
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1answer
43 views

Show that the mapping $(C^1, ||.||_{C^1}) \rightarrow (C, ||.||_\infty) $ is continuous

Assume $D:(C^1, ||.||_{C^1}) \rightarrow (C, ||.||_{\infty}),$ $$D(f) = f',$$ is a mapping with $$||f||_{C^1} := ||f||_{\infty} + ||f'||_{\infty},$$ $$||f||_{\infty} := sup_{x \in [a, b]} f(x).$...
2
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1answer
64 views

$f:[0,1] \to [0,1]$ be continuous bijection , $g \in C[0,1]$ and such that $\int_0^1g(x)(f(x))^{6n}dx=0, \forall n\ge 0$ , then $g=0$?

Let $f:[0,1] \to [0,1]$ be continuous bijection , $g:[0,1] \to \mathbb R$ be continuous such that $\int_0^1g(x)(f(x))^{6n}dx=0, \forall n\ge 0$ , then is it true that $g(x)=0,\forall x \in [0,1]$ ? ...
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0answers
19 views

Continuous Functions with graphing [closed]

I was working on the following problem: Show $f(x)$ is a nowhere continuous function whose absolute value is everywhere continuous $$f(x) = \begin{cases}1 & x \in \mathbb{Q}\\ -1 & x \...
3
votes
3answers
268 views

Are derivatives always continuous? [duplicate]

I am assuming first off that the derivative exists everywhere on the real number line (or everywhere in whatever set you choose to work in if for some insane reason you drag complex numbers or ...
1
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1answer
60 views

Corollary of Tietze extension theorem

The Tietze extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $g:...
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0answers
81 views

Continuity of $f(x, y)=\frac{xy}{\sqrt{|x|} +y^2}$ at $(0,0)$

Assume that $f: \Bbb R^2 \rightarrow \Bbb R$ is defined by $f(0,0)=0$ and, for every $(x,y)\ne(0,0)$, $$f(x, y)={xy \over {\sqrt{|x|} + y^2}}.$$ I have to check whether the function is continuous ...
6
votes
2answers
106 views

$f$ non-constant on $\mathbb R$ such that for any metric $d$ on $\mathbb R$ , $f:(\mathbb R,d)\to (\mathbb R,d)$ is continuous , is $f$ identity?

Let $f:\mathbb R \to \mathbb R$ be a non-constant function such that for any metric $d$ on $\mathbb R$ , $f:(\mathbb R,d)\to (\mathbb R,d)$ is continuous , then is $f$ the identity function i.e. $f(x)=...
4
votes
1answer
59 views

To characterize uncountable sets on which there exists a metric which makes the space connected

For which uncountable sets $X$ is it true that there exist a metric $d$ on $X$ such that $(X,d)$ is connected ? [ The motivation for this question is : I wanted to characterize function $f : X \to X$...
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4answers
109 views

What am I supposed to prove here actually?

I got problems understanding the task: Prove that there are at least two different $x \in (0, 2)$ such that $x^3-x-\sqrt{x} +\frac{1}{2}= 0$. The given hint is to use the intermediate value ...
0
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1answer
12 views

An increasing function defined on a interval that is only continuous outside a countable set

Let $C$ be a countable subset of $(a,b)$. Then there is an increasing continuous function on $(a,b)$ that is continuous only on $(a,b)\setminus C$ This is an example from Royden's real analysis book....
1
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1answer
29 views

Continuity of $f_\alpha(x,y)=\frac{xy}{(x^2+y^2)^\alpha}, f_\alpha(0,0)=0$ in $(0,0).$

I've already proved that $f_\alpha$ is discontinuous if $\alpha\geq 1$. Now I want to prove (what I assume, but don't know) that $f_\alpha$ is continuous if $\alpha<1$. The definitions of ...
0
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1answer
24 views

Does function that maps bounded convex sets (minus straight line segments) to bounded convex sets must be continuous everywhere?

This question in the title came to my mind while I was sitting with my granny in front of my house maybe about half an hour ago. Although it looks innocent I do not know at the moment some simple ...
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3answers
56 views

$f:S^1 \to \mathbb R$ be continuous , is the set $\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ infinite ?

Let $f:S^1 \to \mathbb R$ be a continuous function , I know that $\exists y \in S^1 : f(y)=f(-y)$ where $y \ne -y $ (since $||y||=1$) , so that the set $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=...
3
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1answer
42 views

$f \in C [0,1]$ , $\lim _{x \to 0+} f(x)/x$ exists finitely , $\lim_{n \to \infty} n\Big(n \int_0^1 f(x^n)dx-\int_0^1 \dfrac {f(x)}x dx\Big)=$? [closed]

Let $f:[0,1] \to \mathbb R$ be a continuous function such that $\lim _{x \to 0+} \dfrac {f(x)}x$ exists finitely . Then does the limit $\lim_{n \to \infty} n\Big(n \int_0^1 f(x^n)dx-\int_0^1 \dfrac {f(...
0
votes
1answer
91 views

Existence of metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is everywhere discontinuous

Does there exist a metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is everywhere discontinuous ? It is motivated from this question which ...
1
vote
1answer
28 views

Prove that $f(x, y, z) = x^2+y^2+z^2+2x+2y+2z+3$ is a continuous function from $R^3$ to $R$. (i.e. show that $f^{-1}$ ((a, b)) is open in $R^3$ .)

I'll start by stating I have found a very similar question already posted, but that the solution the asker has accepted isn't helping me understand what I need to do Prove that a function is ...
1
vote
1answer
56 views

Prove that an arbitrary norm is continuous. Is my proof correct?

Let $f: \mathbb{F}^n\rightarrow \mathbb{R}$ be defined by $f(a_1,\cdots, a_n)=\|\sum a_jv_j\|$. Show $f$ is continuous on $\mathbb{F}^n$. 1. $\|\cdot\|$ is an arbitrary norm on $\mathcal{V}$. ...
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0answers
89 views

Continuous embeddings

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ (the ...
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0answers
55 views

Continuity of the period of solutions of a second order ODE, with respect to their initial conditions

There is a second order ODE $$\ddot{x} + b(x) \dot{x}^2 + c(x) = 0$$ with continuous, and locally lipschitz coefficients b, c : $\mathbb{R}\to\mathbb{R}$. Assume the ODE has 2 partial periodic ...
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2answers
52 views

Continuous functions in Topologies

I'm having quite some difficulty finding continuous functions between topologies. Find a continuous function $f:\Bbb{R}_{cocountable} \rightarrow \Bbb{R}_{ususal}$ I'm not sure maybe something that ...
3
votes
2answers
83 views

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is not continuous?

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is not continuous?
1
vote
1answer
35 views

$\epsilon - \delta$ definition to prove that this graph is not continuous at $a$

It is a fundamental problem, and there are some related problem asked before: 1. $\epsilon - \delta$ definition to prove that f is a continuous function. 2. How to show that $f(x)=x^2$ is continuous ...
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0answers
120 views

Problem regarding continuous embeddings [duplicate]

Given the following exercise: We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \{x \in (\mathbb R_+) : f(x) \neq 0\} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ and for all $f \in C^...
0
votes
1answer
58 views

Absolute continuity and continuity

Suppose we have a measure $\mu$ on $(a,b]$ such that $\mu(a,b]=F(b)-F(a)$ where $F$ is non-decreasing, continuous function from the right, Definition: A function $F$ is said to be absolutely ...
1
vote
2answers
28 views

Continuous functions in a topology

The projection function $f : R_{usual}^2 \rightarrow R_{usual}$ given by f(x, y) = x is continuous. Can someone please provide a proof to this, in general when you need to show a function is ...
0
votes
1answer
42 views

Urysohn's extension theorem

Currently I am working my way through Ernest Michael's first article on continuous selections. Here, Urysohn's extension theorem is stated as follows: For a $T_1$-space, the following properties ...
3
votes
2answers
43 views

Compact Sets of $(X,d)$ with discrete metric

Let $X \neq \emptyset$. Define the discrete metric on $X$ with: $ d(x,y)=\left\{\begin{array}{ll} 1, & x \neq y \\ 0, & x=y\end{array}\right.$ (a) Ascertain the compact ...
0
votes
2answers
47 views

Show that $f(x) = \inf\{d(a,x) : a \in A \}$ is continuous

Let $A$ be a non-empty set in a metric space $(X,d)$. Define $f: X \to \mathbb{R}$ by $f(x) = \inf \{d(a,x) : a \in A \}$. Prove that $f$ is continuous. If $f$ is continuous, then $\forall \epsilon &...
0
votes
0answers
17 views

Showing $(C[0,1],d_{\infty})$ is connected

Prove that the metric space $(C[0,1], d_{\infty})$ is connected. Is it path connected? I know how to typically show that a set is connected, but to show $C[0,1]$ is connected is currently escaping ...
1
vote
1answer
26 views

Holder continuous but not Lipschitz

Is there a function that is Holder continuous but not Lipschitz continuous?
1
vote
2answers
67 views

Showing the continuity of $d(x,f(x))$

Assume that $(X,d)$ is compact, and that $f: X \to X$ is continuous. Show that the function $g(x) = d(x,f(x))$ is continuous and has a minimum point. Consider the function $g(x) = d(x,f(x))$. If $g$ ...
1
vote
3answers
71 views

How is the function $f: \mathbb{Z} \to \mathbb{R}$ continuous?

Where $\mathbb{Z}$ is the set of integers and $\mathbb{R}$ the set of real numbers. In a question in a problem sheet, it said this statement was correct, however I do not understand how. You ...
1
vote
4answers
58 views

The continuity of function's restrictions implies the continuity of function.

Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove ...