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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$f:\mathbb R \to \mathbb R$ is continuous and lim$_{n\to \infty} f(nx)=0$ for all real $x$ $\implies $ lim$_{x \to \infty}f(x)=0$

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that for all real $x$ , lim$_{n\to \infty} f(nx)=0$ , then how to prove that lim$_{x \to \infty}f(x)=0$ ? Please help and please don't ...
0
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1answer
19 views

Define multiple-variable function to be continuous

Define the function $f(x,y)= {{x^2 + y (x^2 + y)} \over {x^2 + y^2}}$ at $[0,0]$ so that the function would be continuous. I need help with this calculus problem. I mean, I guess it involves some ...
2
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1answer
66 views

Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$

Let $f\in C^{\infty} ([-1,1])$ and suppose there's a constant $M>0$ such that $|f^{(j)}(x)|\le M:\forall j\in\mathbb N$ (including zero) and for all $x\in [-1,1]$. Prove that if $f(\frac 1 ...
0
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0answers
26 views

Continuity and differentiability of a function. [duplicate]

How to prove that $f(x)$ is discontinuous at $x=0$ ? $$f(x)=\begin{cases} \sin\left(\tfrac1x\right),& \text{when $x\neq0$} \\ 0, & \text{when $x=0$} \end{cases}.$$
3
votes
1answer
25 views

Show for whih values this following function is continuous

For the function $f: [0,2 \pi] \rightarrow \mathbb{R}$ ,state at which points $c \in [0, \pi]$ is $f$ continuous or discontinuous. $$f(x)=\begin{array}{cc} ( & \begin{array}{cc} ...
2
votes
2answers
197 views

Proof that if f is function, continuous on an interval I then f(I) is also an interval

The theorem would be: Let $f:E\to\mathbb{R}$ a continuous function and $I$ and interval, $ I \subseteq E $. Then $f(I)$ is also an interval. I'm not sure if I've understood completely what I have to ...
0
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1answer
32 views

Continuity proof of two-variable function.

The Assignment Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} ...
0
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1answer
38 views

Determine if the following function is continuous in $(0,0)$.

Assignment: Determine if the following function is continuous in $(0,0)$. $$f: \mathbb{R}^2 \rightarrow \mathbb{R},\begin{pmatrix}x\\y\\\end{pmatrix} \rightarrow \begin{cases} 1& ,x≤ 0, y ...
2
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1answer
45 views

How to prove that $f_n(x) = \frac{1}{1+n^{2}x^{2}}$ is continuous on $[0,1]$?

I am having trouble verifying continuity. This seems like a very simple problem but I am not sure if my approach is correct: To prove that $f_n(x) = \frac{1}{1+n^{2}x^{2}}$ is continuous on $[0,1]$, ...
0
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2answers
65 views

$f :\mathbb R\to \mathbb R$ is a continuous function of period $1$ , then $f$ is uniformly continuous on $\mathbb R$

Let $f :\mathbb R\to \mathbb R$ be a continuous function such that $f(x+1)=f(x) , \forall x\in \mathbb R$ i.e. $f$ is of period $1$ , then how to prove that $f$ is uniformly continuous on $\mathbb R$ ...
1
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1answer
44 views

$V$ is open , then $V=\{x\in \mathbb R:f(x)>0\}$ for some continuous function $f$

Let $V$ be a non-empty open set of real numbers , then how to prove that there is a continuous function $f:\mathbb R\to \mathbb R$ such that $V=\{x\in \mathbb R:f(x)>0\}$
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3answers
94 views

Proving continuity by epsilon-delta proof for a function of two variables.

On account of a SE question , I raised the following question. Let $f:D \to \mathbb R^2$ be a function in two variables. How would we go about setting up an epsilon-delta proof? Let $f$ for example ...
0
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0answers
59 views

Prove a two variable function to be continuos on an specific domain

Let the function $g:D \to \mathbb R^2$ be given by $$g(x,y)=2-|x+y|$$ with domain $D= \{(x,y) \in \mathbb R^2:x+y \leq2\}.$ How to prove it is continuous? I know that I need to prove this for every ...
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3answers
28 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
1
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3answers
97 views

Continuous map from the ring on the unit circle

Is there a surjective continuous map from the ring $r<x^2+y^2<1\,(0<r<1)$ on the unit circle $x^2+y^2<1$ ? It seems NO, but how can it be done ? Edit: what if we add the ...
0
votes
2answers
36 views

$f$ is continuous and $f(V)$ is open whenever $V$ is open $\implies$ $f$ is monotone

Let $ A $ be a non-empty subset of $\mathbb R$ and $f : A \to \mathbb R$ be a continuous function on $A$ such that $f(V)$ is an open set for any open set $V$ , then how to prove that $f$ is monotone ? ...
0
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1answer
30 views

Proving a function is not uniformly continuous.

I am using the definition: $(∃ε > 0)(∀n ∈ N)(∃ x_n, y_n ∈ (0,1])[(|x_n − y_n| < δ_n =1/n) ∧ (|f(x_n) − f(y_n)| ≥ ε)]$ to prove that $1/x^2$ is not uniformly continuous. In the solution I am ...
2
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2answers
66 views

Constructing a function similar to x^3 between [0,1]

I'm trying to construct a function $f$, in order to normalize a dataset(obviously where all the element come from $[0,1] \in \mathbb{R}$. The big picture is that the envisioned $f: [0,1] \rightarrow ...
0
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1answer
49 views

Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...
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0answers
27 views

Show that $f: K_1(0) \rightarrow \mathbb{R}^3$ is Lipschitz.

Firstly, the Assignment: Let $V = (\mathbb{R}^3 ,\|\cdot\|_{\infty})$ where $\|\cdot\|_{\infty}$ denotes the maximum norm and consider the function: $$f: K_1(0) \rightarrow ...
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0answers
7 views

Feasability of infinite number of linear inequalities

Consider a continuous function $f:A\to\mathbb{R}^N$ for a closed interval $A\subset \mathbb{R}$. Are there suffieint or necessary conditions for the existence of a solution $w\in\mathbb{R}^N$ such ...
0
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1answer
71 views

Continuous Function for 3 points

Let $f : \mathbb{R} → \mathbb{R}$ be a function with all fibres $(\lbrace{x ∈ \mathbb{R}| f(x) = c\rbrace} = f^{−1}(c), c ∈ \mathbb{R})$ either empty or consisting of exactly three points. Find a ...
2
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1answer
26 views

Relation between continuity and weak star continuity

Let us have a mapping $T:X^*\to X^*$. We can endow domain and codomain with norm 'strong' topology (let X be Banach space), or weak star topology. That gives us total 4 combinations: weak*-weak* ...
0
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1answer
22 views

Continuity of Multiplication by Fixed Element

This is likely a simple question that I'm just missing, but nothing immediately came to mind. When dealing with topological monoids, it is necessary to prove that the group operation is continuous. ...
5
votes
2answers
93 views

$\{a$ : $\forall f\in C^0$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)\}$

Determine all $a\in[0,1]$ such that for ${\it every}$ continuous function $f:[0,1]\to \Re$ with $f(0)=f(1)$ there exists at least one $x$ where $f(x) = f(x+a)$. Firstly, $a=0,1/2,1$ are obviously ...
1
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1answer
37 views

Replacing both f(a) and f(b) with f(c) (confused, please help)

I came across a step in a proof which is puzzling me. The step basically makes the following claim: If $f$ is continuous on some interval $I$, and $a,b \in I$, then there exists $c \in I$ such that ...
2
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1answer
60 views

Topological definition of continuity and its application to epsilon-delta definition?

So I am beginning Munkres' textbook on topology. The topological definition of continuity reads: $f:X\rightarrow Y$ is continuous if for each open subset $V\subset Y$, $f^{-1}(V)$ is an open subset ...
2
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1answer
68 views

Monotonic function satisfying darboux property $\Rightarrow$ continuous

Assume $f : I \rightarrow \mathbb{R}$ is a non-decreasing on an open interval $I$ and that $f$ satisfies the Intermediate value property or Darboux's property on $I$ (that is, for any $a < b$ ...
15
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3answers
315 views

Continuity of a function in two variables

Function $f(x,y)$ is continuous in each variable separately. Prove that there exists a point where it is continuous in two variables. I do not quite understand how to act here. I know the ...
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2answers
31 views

Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
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0answers
27 views

Questions from a calculus assignment about a function [duplicate]

Can anyone guide me through this problem? Let $f(x) = \lvert 4-x^2 \rvert$, $-4\le x\le 1$. Sketch (I have completed this part). Rewrite $f$ as a piecewise function. Give the range of $f(x)$. Use ...
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0answers
18 views

Continuity of a piecewise defined function in two variables

I need some insight into the “approach” that I used to solve this problem. Namely, I was asked to find if the following function is continuous on all $\mathbb{R}^2$: $$ f(x, y) = \left\{ ...
1
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4answers
55 views

f is continuous at x = a

Let $f$ be a continuous function defined on $\mathbb{R}$ In case of $f(0)=-1$ Prove that there exists values $x>0$ with $f(x)<0$ In case of $f(1)=1$ Prove that there exists values ...
4
votes
3answers
82 views

If $f$ is continuous then there exists $x\in [0,1]: f(x)=x$

I wish to prove the following by contradiction: Let $f:[0,1]\rightarrow[0,1]$ be a continuous function. Prove that there exists $x\in [0,1]$ such that $f(x)=x$. Proving this directly, one would ...
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1answer
24 views

Condition for continuity of bilinear form

In my numeric script there is a unproved theorem, saying that a bilinear form $a \colon V\times V \to \mathbb{R}$ on a normed vector space $V$ is continuous if and only if $$|a(v,w)| \leq c \, \|v\| ...
3
votes
2answers
33 views

How to “convert” from net to sequence in a first countable space

In a first countable space, what's a good way of going from nets to sequences? Let me explain more clearly what I mean. Suppose $f:X\to Y$ is a topological map and $X$ is first countable. Then I ...
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1answer
31 views

Definition of a continuous function

I am struggling to understand a basic definition of a continuous function from a textbook: A function f is continuous if for all x, and for all $\epsilon>0$, there exists $\delta>0$ such that ...
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1answer
30 views

Continuity of a Function

I'm dealing right now with properties of a function and I have to prove if a given function is injective, surjective or bijective. I prove injectivity with the formula $x_1 = x_2 \Rightarrow f(x_1) = ...
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0answers
25 views

extension theorems on normed spaces

I know that there are a number of extension theorems, Tietze's extension theorem, Hahn-Banach extension and so on.. I want to know if there is an extension theorem which guarantees that if say $X$ is ...
1
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1answer
60 views

Proof that eigenvector corresponding to simple eigenvalue is continuous

Let $\lambda$ be a simple eigenvalue of $A \in L(C^n)$ and let $x$ be the corresponding eigenvector. Then for $E \in L(C^n)$, $A+E$ has an eigenvalue $\lambda(E)$ and an eigenvector $x(E)$ such that ...
0
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1answer
15 views

Lower semi-continious, compactum, minimum

Let $M$ be a topological space, $K\subset M$ compact, $f\colon M\to\mathbb{R}\cup\left\{+\infty\right\}$ lower semi-continious. Show that $f$ takes its minimum on $K$. Good day, we ...
0
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1answer
41 views

If $f$ and $g$ are continuous, prove or disprove that the set $\{x \in \mathbb{R} : f(x)\le g(x)\}$ is closed

Let $f,g : (\mathbb{R};J_s)\to (\mathbb{R};J_s)$, (where $J_s$ is the usual (standard) topology on $\mathbb{R}$) be continuous. Prove or disprove: (a) the set $\{x\in \mathbb{R} : f(x)\le g(x)\}$ ...
0
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1answer
38 views

If a differentiable function has bounded derivative, Must it be that its derivative continuous?

I got this question: Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, If $f'$ is bounded on $(a,b)$, Must it be the case that $f'$ is ...
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1answer
99 views

How to prove uniform continuity problem!

A) $f(x)=x^3$ , give an example of an interval where $f$ is uniformly continuous and another where it is not. explain your choose of examples B) decide if $f(x)= \dfrac{1}{\sin x} - \dfrac{1}{x}$ is ...
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1answer
45 views

Is a function continuous iff its restriction to each element of an open cover is continuous

Let $(X;T_1)$ and $(Y;T_2)$ be topological spaces and let $A$ and $B$ be nonempty subsets of $X$ with $A\cup B= X$ Suppose $f:X\rightarrow Y$ is a function. Then prove or disprove: (a) if $f_A$ and ...
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1answer
25 views

Is the Inverse of the Vectorised Solid Angle Equation for $n$ Circular Discs Continuous?

I have a continuous function$^{*1}$ that takes in 3 arguments, and returns 24 outputs. I want to know if the inverse of this function is continuous. The 3 input arguments are the x, y, and z position ...
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0answers
37 views

Proving $\lim_{x\to c} g\circ f(x)= g(b)$ without sequential criterion

Pardon my English beforehand. I want to prove, without using the sequential criterion for continuity, the next theorem: Let $f$,$g$ be defined on $\mathbb R$ and let $c\in \mathbb R$. Suppose ...
1
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0answers
24 views

Does right continuity imply only countably many discontinuities? [duplicate]

Does right continuity imply only countably many discontinuities? That is, if $f:\mathbb{R}\rightarrow \mathbb{R}$ is right continuous then does it only have countable many discontinuities? Thanks
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2answers
32 views

How would I finish this continuity proof?

I have a multivariable function $f$ with $$f(x, y) = \begin{cases} \frac{x^2+y^2}{y} & \text{if }y \neq 0\\ 0 & \text{if }y = 0 \end{cases}$$ and want to show that it is continuous at $(0, ...
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vote
2answers
78 views

How to prove that $f(x,y)=3+2x+y$ is continuous?

The question is to prove that the function $f(x,y,z) = 3+2x+y$ is continuous everywhere. My approach uses the delta-epsilon method. $|(x,y)-(a,b)|\lt \delta$ then $|f(x,y)-f(a,b)|$. All I did was ...