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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Continuity of a Single Point

My problem is :Find the points at which the the mentioned function is continuous $$f(x) = \begin{cases} x & \text{if $x$ is a Rational Number} \\ -x & \text{if $x$ is not a ...
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128 views

$f_n(x) = \cos \left( \sqrt{x^2 + \frac{1}{n} }\right)$ is uniformly convergent

I need to prove that this sequence of functions: $$f_n(x) = \cos \left( \sqrt{x^2 + \frac{1}{n} }\right)$$ converges uniformly on $[0,1]$. This is a question regarding this one. There's an answer ...
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166 views

How to prove $f(x) = \ln x$ continuous by proving first that $f(x)$ continuous at $1$, and then by using $\ln (xy) = \ln(x) + \ln(y)$. [duplicate]

I have a question concerning the proof of the continuity of $f(x) = \ln x$. I read in a comment by Pedro Tamaroff to ncmathsadist's answer to this question that this can be proved in two steps: ...
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34 views

Absolute value and $\varepsilon - \delta$ proofs of continuity

I am seriously struggling with $\varepsilon - \delta$ proofs overall continuity of a function (e.g. here there is an attempt). In the attempt to see how to prove the continuity of $f(x) = x^2 - 2$, I ...
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19 views

If possible calculate the norms of the following linear forms.

Let be $I$ and $d$ defined as $$I:f\to\int_{-1}^{1} f(x)dx \\ d:f\to f(0)$$ on $\mathcal{C}^1[-1,1]$. Check if the linear forms are continous with respektive to the norms $\|\text{ }\|_{\infty}, \|\...
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33 views

$f$ is continuous at $p$ iff $f(p_n) \to f(p)$ for all $p_n \to p$?

Let $f$ be a mapping of topological space $X$ to topological space $Y$. It's straightforward to check that if $f$ is continuous at $p$ we have $f(p_n) \to f(p)$ for all $p_n \to p$. Is the converse ...
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73 views

continuity at isolated point [duplicate]

In our booklet it is written that : $$A\ function\ is\ continuous\ at\ every\ isolated\ point.$$ MY doubt:- Let us consider an example: Let $f:N\rightarrow R$ $\ $ such that $$f(x)=x$$ As $\ N=\...
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240 views

Every continuous and bounded $f:\mathbb{R}\to\mathbb{R}$ such that $f(x) = \int_x^{x+1} f(y) dy$ for every $x$, is constant

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous and bounded function such that for all $x$: $$f(x) = \int_x^{x+1} f(y) dy.$$ Prove that $f$ is constant. I could have progressed if it was given ...
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1answer
62 views

$\varepsilon - \delta$ proof that $f(x) = x^2 - 2$ is continuous - question concerning the initial choice of $\delta$

I just realized I did non really internalized $\varepsilon - \delta$ proofs. Here there is an attempt, with some general questions I have. Proposition: $f(x) := x^2 - 2$ is continuous. ...
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62 views

A discontinuous function on $\mathbb{R}^n$

I have read a paper and I did not understand with the following statement: Let $r_1, r_2,...$ be an enumeration of $\mathbb{Q}^n \subseteq \mathbb{R}^n$. Given functions $g:\mathbb{R} \to \mathbb{R}$ ...
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1answer
65 views

Showing Lipschitz continuity

Let $f: \Bbb R^n \to \Bbb R^m$ be a function with the property that for all $v \in \Bbb R^n$ an $L=L(v) \gt 0$ exists so that for all $x \in \Bbb R^n$ the function $t \longmapsto f(x+tv)$ is $L$-...
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17 views

Let $f:R^n\rightarrow R$ be a lower semi-continuity function, how to show for any constant $r$ , $U=\{z\in R^n : f(z)> r\}$ is open?

Let $f:R^n\rightarrow R$ be a lower semi-continuity function, how to show for any constant $r$ $U=\{z\in R^n : f(z)> r\}$ is open ?
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83 views

Obscure proof that $+$ and $\times$ are continuous?

I am looking for proof of $+$ and $\times$ are continuous operations without using the standard definition of continuity (1. $\epsilon-\delta$, or 2. preimage of open sets or 3. sequential ...
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28 views

Why is $\log z = \ln r + i\theta$ ($r>0, \alpha <\theta < \alpha + 2\pi$) discontinuous at $\alpha$?

In one book on complex variables it is written that, given the function $\log z = \ln r + i\theta$ (for proper citation, let's call it function (2), as in the book) ($r>0, \alpha <\theta < \...
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1answer
29 views

Prove $\lim \sup f (x_n) = f(\lim \sup (x_n)) $ and same for $\inf$

Prove: Let $A \subset \mathbb{R}$ compact, $f: A \rightarrow \mathbb{R}$ continuous, increasing monotone and $(x_n) \subset A$. Consider. Show that $\lim \sup f (x_n) = f(\lim \sup (x_n))$ and $\lim \...
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24 views

How to construct continuous functions on arbitrary topological spaces?

Constructing continuous functions $\mathbb{R} \to \mathbb{R}$ (with the euclidean topology) is easy: there is a quite large collection of elementary functions, and we can get even more by composing/...
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1answer
26 views

Real Analysis & Continuity

If $f:\mathbb{R} \to \mathbb{R}$ is a continuous function such that $f(x)=x$ has no real solution, then show that $f(f(x))=x$ has no real solution either. Is the proof trivial as it seems or does it ...
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110 views

How to show that $f(x) = x^2$ is continuous using topological definition?

I am trying to show that simple continuous functions satisfy topological definition of continuity Recall given $(X, \mathcal{T}), (Y, \mathcal{J}), f$ is continuous if $f^{-1}(V) \in \mathcal{T}, \...
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30 views

Problem in finding the points of discontinuity of two functions.

How can I find the points of discontinuity of the functions : (a) $f(x) = \lfloor x^2 \rfloor \sin (\pi* x)$. (b) $f(x) = \lfloor x \rfloor + (x - \lfloor x \rfloor)^{\lfloor x \rfloor}$. Where, $\...
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38 views

Uniform Continuity

This question has three parts. a) Difference between continuity and uniform continuity b) Geometrical meaning of uniform continuity c) Correct the example Definition of Continuity of a function ...
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50 views

Given a set of conditions for $f$, prove $f$ is continuous $\forall x \in \mathbb{R}$

Problem: Let $f(x)$ be a function whose domain is $\mathbb{R}$. It is known that $f(x)$ is continuous at $0$ $f(x+y) = f(x)f(y)$ $\ \ \forall \ x, y \in \mathbb{R}$ Show that $f(x)...
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36 views

Differentiability of rationals and irrationals of function

First let $q_{j}$ be an enumeration of rationals. Let $f(x)=\sum_{k=1}^{∞} |x-q_{j}|/2^j$ for $x∈(0,1)$ Show that: $f$ is continuous on $(0,1)$ $f$ is not differentiable at the rational points of $...
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65 views

Continous function $f$

I have a function $f:[a,b]\longrightarrow [c,d]$ that is bijective and monotone increasing. I have to show that f is also a continous function. I wanted to show this by contradiction that for f is not ...
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55 views

Find the value of the constant k that makes the function continuous

h(x) = \begin{cases} x^{2} & \text{if $x\le5$} \\ x+k & \text{if $x>\ 5$} \\ \end{cases} Answer choices are A. k=20 B.k=-5 C. k=5 D. k=30
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23 views

C¹ function in compact and polygonal path connected implies Lipschitz

"Let $f: \Omega \rightarrow \mathbb{R}^{m}$ such that $f \in C^{1}(\Omega).$ Show that, for $K \subset \Omega$ compact and polygonal path connected, $f|_K$ is Lipschitz." I can relate the function ...
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118 views

A discontinuous function with smooth sections

I am searching for $f : U\rightarrow \mathbb R $ defined in an open square $U$ in $\mathbb R^2$ so that $(0,0) \in U$, $f$ is not continuous at $(0,0)$, for each $x$ the function $y\mapsto f(x,y)$ is ...
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19 views

A Study of Continuity of a given Function

I am trying to figure out if the following function is in fact continuous: given $$f(x,y) = \left\{\begin{array}{cc} \frac{|y|-|x|}{y^2} & |x| < |y| \\ 0 ...
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25 views

Nonexistence of a continuous injection from $\mathbb{R}^n$ to $\mathbb{R}$, for all $n \geq 2$. [duplicate]

I'm trying to do the following exercise from my lecture notes: There does not exist a continuous injection from $\mathbb{R}^n$ to $\mathbb{R}$, for all $n \geq 2$. I don't really know where to ...
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15 views

Problem with continuity and limits in 3 dimensions

Given the function $\lim_\limits{(x,y)\to (0,0)}$ $\frac{x^2y^2}{x^4+3y^4}$ The website I was reading lecture notes from said this function is not continuous at the point in question but doesn't go ...
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45 views

Prove the function is continious.

If the function $f(x)$ is continious at $x=0$, using definitions show that $f(rx)$ is continious at $x=0$. Here $r$ is a real number.
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1answer
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Showing Intermediate Value property and closed preimage implies continuity

Let $f : [0,1] \to \mathbb{R}$ be a function satisfying the Intermediate Value property. Assume that for any $y \in \mathbb{R},$ the preimage $f^{-1}(\{y\})$ is closed. Prove $f$ is continuous. ...
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21 views

Error term between $f(x)$, its average value and value at midpoint

Let $f$ be a smooth function on interval $[a,b]$. Define the average $\bar{f}=\dfrac{1}{b-a}\int_a^bf(y)\,dy$ and $\bar{x}=\dfrac{a+b}{2}$, then for any $x\in [a,b]$, we can write $$f(x)-\bar{f}=c(x-\...
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1answer
30 views

continuous random variable expectation and variance [closed]

you have a continuous random variable X uniformly distributed, and E(X) = 3, calculate the V(X) am stuck, how am i supposed to get the variance with no function in the first place
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1answer
78 views

Is the mapping “positive stochastic matrix onto its Perron-projection” continuous?

I am dealing with a topological question concerning the mapping that maps a positive stochastic matrix onto its invariant distribution. I am asking myself if such a mapping is continuous (or ...
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39 views

Show that function $f$ is not continuous in $x=0$ for all $c\in\mathbb{R}$

Show by $\varepsilon-\delta$-criterion that for each $c\in\mathbb{R}$, the function $f\colon\mathbb{R}\to\mathbb{R}$, $$ f(x)=\begin{cases}\frac{1}{x}, & x\neq 0\\c, & x=0\end{cases} $$ is not ...
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1answer
69 views

If $f \circ f$ continuous prove $f$ continuous

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ strictly increasing so that $f \circ f$ is continuous. Prove $f$ is continuous. I can prove this using sequences, but it's quite tedious. My question is: ...
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1answer
28 views

Global Lipschitz implies bounded in coefficient

Consider $g:\mathbb{R}^2\to \mathbb{R}$ of the form $g(x,y)=p(x)q(y).$ Assume $g$ is uniformly Lipschitz in $x,y$ in the sense that there exists $K>0$ such that for any $(x_1,y_1),(x_2,y_2)\in \...
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1answer
66 views

Show Lipschitz continuity of a function

I'm stuck trying to solve the following exercise: Let $f:\mathbb R^n \to \mathbb R^m$ a function with the property that, for all $v \in \mathbb R^n$, there is $L=L(v) > 0$ such that the ...
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1answer
53 views

All Continuous function can be drawn? [duplicate]

I googled and came to know that there are many continuous functions which cannot be drawn by hand, like Cantor, Weierstrass functions etc. Now this question was asked in a college admission interview....
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123 views

Where is $x^x$ continuous?

The idea of continuity of a function is something I come across quite regularly, but I've never really understood it well. I'm trying to fix that by looking at some interesting functions. What ...
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2answers
30 views

Number of discontinuous values

We have to find the number of values of $x$ at which the function $$ f(x) = \frac{2x^5-8x^2+11}{x^4+4x^3+8x^2+8x+4}$$ is discontinuous. I thought that since both numerator and denominator are ...
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2answers
60 views

Can continuity of real functions be “globally” characterized?

Most characterizations of pointwise continuous functions defined on an interval rely on "local" properties. That is, a function is continuous at $x_0 \in I$ if it satisfies some property (epsilon-...
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1answer
36 views

Linear function: relation between linearity and continuity

Given a linear function $A$ between two normed Vectorspaces i have to show euquality of the follwing statements: $A$ is continuous There exists a point where $A$ is continuous $A$ is Lipschitz-...
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56 views

Prove or disprove continuity of two maps

Yet another time I need help to prove continuity of a certain map and don't know how to do it: Look at the vector space $$C_b^1(\mathbb R; \mathbb C) := \{f \in C^1(\mathbb R;\mathbb C):||f||_{\...
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37 views

Equivalence of statements about a linear map

I need someone to help me solve the following exercise: Let $(X, ||\cdot||_X)$ and $(Y, ||\cdot||_Y)$ be normed vector spaces over a common field $\mathbb K$ $(\mathbb R$ or $\mathbb C)$. For a ...
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35 views

Is there a math notation/ term for “$f(x_n) \to 0$ iff $g(x_n) \to 0$”?

I have two real-valued functions $f,g$ defined over the $N$-dimension Real Euclidean space: $$ f,g: \mathbb{R}^N\to\mathbb{R}. $$ They satisfy this property: $$ \forall x_n \in \mathbb{R}^N: f(x_n)\to ...
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2answers
43 views

How do I examine f on continuity?

Let $f$ be defined as follows: $$f:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto\begin{cases}\frac{xy^{2}}{x^{2}+y^{4}}&\text{if } (x,y)\neq (0,0)\\ 0&\text{if } (x,y)=(0,0)\end{cases}$$ How do I ...
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14 views

Checking of uniformly continuity of the following functions

Which of the following 4 functions are uniformly continuous? and which are not? I want to know the process/explanation of the solutions.
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30 views

Proof/disprove contunuity of a map [duplicate]

I need help with proving / disproving something: Look at the map $$\Phi: (C([0,1], \mathbb R), ||\cdot||_{\infty}) \to (\mathbb R, |\cdot|); \,\,\,\,\,\,\Phi(u) := \int_0^1 u²(t) dt$$ a) ...
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1answer
47 views

Is the given function $f$ continuous?

Problem Let $\mathbb{R}_l$ denote the reals with lower limit topology, and let $\mathbb{R}_l\times \mathbb{R}_l$ have the product topology. Then the map $f:\mathbb{R}_l\times\mathbb{R}_l\to\mathbb{R}...