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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4answers
88 views

Assume that $ f: R \to R $ is uniformly continuous. prove that there are constants A,B suchthat $ |f(x)| \le A + B|x| $ for all $ x \in R $.

Assume that $ f: \mathbb R \to \mathbb R $ is uniformly continuous. prove that there are constants $A,B$ such that $ |f(x)| \le A + B|x| $ for all $ x \in \mathbb R $. my concern is just $f$ is ...
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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 ...
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1answer
54 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 $\...
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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 $c\in \mathbb R\setminus \mathbb Q$. ...
1
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2answers
45 views

Compactness of the set of points where a continuous function achieves a local maximum

Let $(K,d)$ be a compact metric space, and $f:K\rightarrow \mathbb{R}$ be a continuous function on $K$. Define: $$M=\left \{ x\in K :\text{$f$ achieves a local maximum in $x$} \right \}$$ I need to ...
7
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1answer
115 views
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Differentiablity at $0$ of a function $f: \mathbb R \to \mathbb R$ which is twice differentiable in $\mathbb R \setminus \{0\}$

Let $f: \mathbb R \to \mathbb R$ be a function , twice differentiable in $\mathbb R \setminus \{0\}$ such that $f'(x)<0<f''(x) , \forall x <0$ and $f'(x)>0>f''(x) , \forall x >0$ ; ...
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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 $R^D$ ...
3
votes
1answer
87 views

Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$

I have the following exercise: $$f:[0, +\infty) \rightarrow \mathbb{R} \text{ uniformly continuous } .$$ $$\text{Show that there are } a,b \geq 0 \text{ so that } |f(x)| \leq ax+b, \forall x \geq 0.$$...
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 ...
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+50

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
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1answer
282 views

Discontinuity of floor function

I am still getting confused with showing discontinuity of functions, here is my attempt at a question , if someone could check to see if I am going about it in the correct way. $f: \mathbb{R} \...
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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 ...
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....
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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: ...
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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 ...
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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 ...
2
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1answer
108 views

Another functional equation: $f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor$

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
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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 ...
1
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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^+}...
3
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3answers
124 views

Continuity of $\frac{x^3y^2}{x^4+y^4}$ at $(0,0)$? [duplicate]

Suppose a function $f$ is defined as follows: $$f(x,y)=\begin{cases} \frac{x^3y^2}{x^4+y^4}&\text{ when }(x,y)\neq(0,0),\\0 & \text{ when }(x,y)=(0,0).\end{cases}$$ Is this function ...
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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: ...
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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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 ...
3
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1answer
92 views

Prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous, then it is continuous from the right

I'm trying to prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous (where the topology of $\mathbb{R}$ is $\emptyset$, $\mathbb{R}$, and all sets of the form $(-\infty, a)$), then it is continuous ...
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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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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 $...
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
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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1answer
55 views

Prove that there is no continuous surjection from $S^n$ to $\mathbb{R}_n$

Let $S^n = \{(x_1, . . . , x_{n+1}) \in \mathbb{R}^{n+1} \mid \sum_{k=1}^{n+1} x_k^2 = 1\}$. Prove that there is no continuous surjection $f : S^n \to \mathbb{R}^n$.
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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:...
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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 $...
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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
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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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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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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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35 views

Continuous function with support continuously embedded [duplicate]

Can someone give me a solution for this? We define $$C^1_c(\mathbb R_+) := \{f \in C^1(\mathbb R_+): \overline{\{x \in (\mathbb R_+) : f(x) \neq 0\}} \text{ compact in } (\mathbb R_+,|\cdot|)\}$$ ...
3
votes
1answer
253 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
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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-...
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 $\varphi: I \to \mathbb{R}^n$ ...
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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)\} $ ?
1
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1answer
72 views

In a Completely regular $T_1$ space, two disjoint sets, one compact, the other closed, can be separated by a continuous function?

Let $X$ be a completely regular $T_1$ space and let $A,B$ be disjoint closed subsets of $X$, where $A$ is compact also. Then is it true that there exist a continuous function $f\colon X \to [0,1]$ ...
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
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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 ...
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\...