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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A form for a piecewise continuous function?

Let A be a continuous function, let B be a piecewise constant function, and let C be a multivariate continuous function. Is it true that the parameterization $D(x) = C(A(x),B(x))$ defines all ...
5
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2answers
69 views

Increasing function with $f'(x)=f(f(x))$

Is there a strictly increasing function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x$?
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44 views

if $M$ is compact, then every continuous bijection $F:M\to N$ is an homeomorphism

My book proves that: if $M$ is compact, then every continuous bijection $f:M\to N$ is an homeomorphism by the following: Being $f$ closed, your inverse $g:N\to M$ is a function such that $F\subset ...
1
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1answer
53 views

Continuity of $F(x,y)=|x-y|$

Suppose that $F:\mathbb{R}^2\to \mathbb{R}$ defined by $F(x,y)=|x-y|$. Prove using $\epsilon-\delta$ that $F(x,y)$ is continuous. Let $(x_0,y_0)\in \mathbb{R}^2$. We have to show that for any $\...
2
votes
1answer
27 views

Confusion in finding left and right hand limits [duplicate]

Let $f:\mathbb R$→$\mathbb R$ defined as - $f(x)=0$, if $x$ is irrational or $x=0$ and $f(x)=1/q$, if $x=p/q$, $p\in$$\mathbb Z$ ,$q\in$$\mathbb N$, $(p,q)=1$. What are the points of continuity of $...
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1answer
29 views

Question involving continuity of function

Problem: Function $f$ is defined: $f(x)=x^2$ for $x\in \mathbb Q$ and $f(x)=x$ for irrational $x$. I have to check continuity of function. My work: Let $c\in \mathbb R\setminus \mathbb Q$. ...
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0answers
20 views

Generalization of Strict Local Maxima

I try to generalize a strict local maximum to a local roof which can possibly be a flat area instead of just a single point. Below is my attempt: Let $f$ be a continuous real-valued function on $R^D$ ...
0
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2answers
70 views

Is f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}} $with f(0,0)=0 continuous in (0,0) [duplicate]

I believe that the function: f(x,y)=$\frac{x^{2}y}{x^{2}+y^{4}}$ is continuous on the point (0,0) but i can't prove it. I know you have to choose something like $x=cy^{2}$(with c a constant) to prove ...
0
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0answers
39 views

Work required to align pieces in a plane.

Given two piecewise continuous functions f(x) and g(x) and that $\lim_{a -> x^-} g(a) - f(a) = \lim_{a -> x^+} g(a) - f(a)$ at all points, find the work used to shift each of the planar slolids ...
0
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0answers
32 views

Continous functions and zeros

How to prove following theorem? If sequence $\{f_n\}$ of continous real functions with domain $D \subset \mathbb{R}$ is compact convergent to $f$ and sequence $\{x_n\}$ with $D$ satisfies $f_n(x_n) = ...
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1answer
25 views

Proving the set $E(f) = \{(x,y,z)\in\mathbb{R}^3 \ | \ z > f(x,y) \}$ is open if $f$ is continuous.

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a continous function. I want to prove the set $E(f)$ given by $E(f) = \{(x,y,z)\in\mathbb{R}^3 \ | \ z > f(x,y) \}$ is open. What I have tried so far: ...
1
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2answers
56 views

When to rationalize to repair continuity, and why does it work?

I was working on a question out a GRE math prep book: "Find the inverse of $f(x) = \frac{x}{1-x^2}$ that works for all $x \in \mathbb{R}$ where $f$ is defined over $(-1,1)$" (works meaning is well ...
2
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1answer
49 views

Continuous function on the unit sphere [duplicate]

Let S$^2$ := $\lbrace$ x $\in$ $\mathbb{R}$$^3$ : $\Vert x\Vert$$_2$ $\rbrace$ $\subset$ ($\mathbb{R}$$^3$, $\Vert .\Vert$$_2$) and T: S$^2$ $\to$ ($\mathbb{R}$, $\vert x\vert$ ) a continuous function....
4
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2answers
37 views

Prove if $f(x)$ and $g(x)$ is continuous, then $f(x) + g(x)$ is also continuous using the $\epsilon - \delta$ definition of limits

Since both $f(x)$ and $g(x)$ is continuous, then $$(\forall \epsilon_1 >0)(\exists \delta_1 >0) [\vert x-a\vert< \delta_1 \to \vert f(x)-f(a) \vert <\epsilon_1]$$$$(\forall \epsilon_2 >...
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0answers
33 views

Function from space of continuous functions to reals is continuous (Proof Verification)

Question: $C$ is the space of continuous functions from $[0,1]$ to $\mathbb{R}$ under the sup metric. Prove the function $$f:C\to\mathbb{R}\quad f\to \int_0^1 f(t)^2 dt$$ is continuous. My answer: ...
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1answer
30 views

Bounding a $C^0$ function with $C^1$ functions

Given a continuous function from $(0,+\infty)$ in itself, with $\lim_{x\to 0^+} f(x)=0$, find $C^1$ functions $g,h:(0,+\infty)\longrightarrow(0,+\infty)$ such that $g\leq f\leq h$ and $\lim_{x\to 0^+}...
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1answer
28 views

Removing dicontinuity from functions involving modulo?

I am currently looking into removing discontinuity from piecewise continuous functions without changing the derivative where it is defined and (preferably) the value of right sided limit at 0. This is ...
0
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1answer
25 views

Continuous and differential inverse function

I have a very interesting question: Given a function $f$ which is continuous but need not be differentiable. Then the correct statement is a. it can be an odd function b. it can't be an ...
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votes
2answers
41 views

Real analysis: Continuity and Differentiability [closed]

Let $f(x)=x^2$ if $x$ is rational and $f(x)=0$ if $x$ is irrational. a) Prove that f is continuous at exactly one point, namely $x=0$. b) Prove that f is differentiable at exactly one point, namely $...
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1answer
57 views

Continuous map on $S^2$

Can you help me with this? Let $S^2 := \{x\in \mathbb R^3:||x||_2 = 1\} \subset (\mathbb R^3, ||\cdot||_2)$ and $T:S^2 \to (\mathbb R, |\cdot|)$ be a continuous map. a) Why does T assume its ...
0
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1answer
81 views

Continuous injection and density in $l_p$ spaces

If $r \le s$ then $l_r$$\subseteq$ $l_s$ . How can I prove there is a continuous injection $l_r$ $\hookrightarrow$ $l_s$? The suggestion was to use the fact that $\Vert$x$\Vert$$_r$ $\le$ $\Vert$x$\...
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0answers
26 views

continuity on the given interval

If a function is continuous on $[a,b]$ then it's continous on all points of $(a,b)$. But is vice-versa true.if not then how to do the following problem .Discuss the continuity of $x-|x-x^2|$.without ...
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0answers
7 views

Phase correlation vs. normalized cross-correlation

In 2-dimensional discrete signal analysis (specifically image processing), a definition I found for the normalized cross-correlation between two images, both of size MxN $g_1(x, y)$ and $g_2(x, y)$ is:...
0
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1answer
32 views

existence of certain function on unit interval

I'm trying to solve this exercise in an introductory book on general topology: Let $(X,d)$ be a metric space and $A,B \subset X$ disjoint closed subsets. Show that there exists a continuous function $...
0
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1answer
80 views

Removing jump discontinuity from a tricky function.

I have the function $\cos(x)\lfloor x \rfloor$ which I would like to make continuous without changing the derivative where it exists or the values approaching 0 from the right side. I can do this by ...
0
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0answers
32 views

Function sequence and some properties

Consider functions $f_n$, $f : \mathbb{R} \rightarrow \mathbb{R}$ such that the sequence $\{f_n\}$ is uniformly convergent to $f$ and every $f_n$ has property $W$. Determine whether $f$ must have $W$ ...
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1answer
25 views

Piecewise Functions

I have been working on problems "a" and "b" for the longest time. I know part "a" is not continuous because if I were to draw the graph I would have to pick up the pencil to draw the graph. I don't ...
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1answer
45 views

How to calculate $\lim_{x \to \infty}{\frac{1}{x}\int^{3x}_{x/3}} g(t) dt$?

Function $g: (0; +\infty) \rightarrow \mathbb{R}$ is unbounded, continous and has limit in $+\infty$ equal to $\pi$. How to calculate $$\lim_{x \to \infty}{\frac{1}{x}\int^{3x}_{x/3}} g(t)\, dt?$$
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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
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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
695 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
649 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
40 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
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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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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 $\...
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2answers
41 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\...
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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
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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 ...
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2answers
75 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
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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 ...
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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
votes
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 $g(x)=0,\forall x \in [0,1]$ ? ...
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
267 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 ...