Tagged Questions

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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2
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2answers
45 views

Continuity of functions and rational numbers

I saw this argument used in a proof, and I don't quite understand it with the $\epsilon - \delta$ definition (it is easier to understand with the sequential criterion). If $f: \mathbb{R} \rightarrow ...
1
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2answers
52 views

Where is the following function continuous?

I would like to show where the following function is continuous. $$f(x)=\begin{cases}13&\text{if }\;0\le x\leq 1\\x&\text{if }\;1<x\le 2\end{cases}$$ I am pretty sure its continuous on ...
4
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2answers
184 views

Why “a function continuous at only one point” is not an oxymoron?

I understand that there are functions that, by definition of continuity, can be continuous at only one point, such as $$f(x)=\begin{cases} x,&\text{if }x\in\Bbb Q\\ 0,&\text{if }x\in\Bbb ...
0
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1answer
37 views

is it important to use $(\epsilon - \delta )$ to provide limit in providing continuity of a function at one point

my question seem dull but I really want to know in providing if the f(x) continuous or differentiable at some point a . We must provide that $$\lim_{x\to a}f(x) = f(a)$$ I want to know in this step. ...
1
vote
1answer
18 views

Continuity argument to show that the derivative exists everywhere.

I have shown that, for $f(X) = \det(X)$, $$\mathrm d f_A(H) = \mathrm{tr} (\mathrm{adj}(A) H)$$ But I have only show this for invertible $A$. I wish to use a continuity argument to show that this ...
0
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3answers
24 views

extending definition of a function to make it continuous

I have two functions on hand, namely $$f(x) = x^3\cos\left(\frac1x\right)$$ and $$f(x) = \frac{1-\cos(x)}{x^2}$$ I would like to know where are these functions discontinuous, and how do I extend the ...
0
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0answers
14 views

Right continuity of right inverse of right continous map

I am stuck on the following proof that I found in Dellacherie-Meyer's book "Probabilities and potential B", p. 119 (increasing processes and projectors). Given a map $a$ on $[0,\infty [$ which is ...
0
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1answer
18 views

$C([a,b] \times [c,d],X)$ compared to $C([a,b],C([c,d],X))$ and $C([c,d],C([a,b],X))$

Let $C(Y,X)$ be the space of continuous functions from $Y$ to $X$ together with the supremums norm. Here $Y$ is a compact space and $X$ a metric space. Let $a,b,c,d \in \mathbb R$ be finite, with ...
1
vote
1answer
31 views

Finding all continuous and discontinuous points of composite functions

Let $f(x) = \operatorname{sgn}(x)$ and $g(x) = 1 + x^2$. How do I go about finding all the continuous and discontinuous points of the functions $f\circ g$ and $g\circ f$ ?
0
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3answers
63 views

Prove that the function is continuous at n where n is an integer, but discontinuous elsewhere.

I'm working on my self study again, and I'm given a function $f(x)=\sin\pi x$ , where $x$ is rational and $f(x) = 0$ when $x$ is irrational. How do I prove that the function is continuous at $n$, ...
1
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2answers
38 views

Delta-Epsilon Proof of Continuity of a Function

Define $f\colon \mathbb{R} \times \mathbb{R}\to\mathbb{R}$ as $\dfrac{xy}{x^2 + y^2}$ for $(x, y) \neq (0, 0)$ and set $f(0, 0) = 0$. Determine whether $f$ is continuous. Please keep in mind that I'm ...
0
votes
1answer
20 views

How to prove continuity of addition over weird metric? Edit: Ignore this. Errors in the problem definition.

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
4
votes
3answers
48 views

Limit theorems, prove function has a limit at every point

Suppose that $f:R\to R$ is a function such that $f(x+y)=f(x)+f(y)$ for all $x,y∈R$. Assume that $f$ has a limit at $0$, $f(1)=1$. Prove that $f(x)=x$ for all $x \in R$ Hint: Show first that $f$ is ...
1
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3answers
56 views

If $f(1) > p >1$ then $f$ is increasing.

Consider the function $f$ with the following properties: $$\lim_{x\rightarrow 0} f(x) =1,$$ $$f(x+y)=f(x)\,f(y),$$ $$f(x) >0,\quad \forall x\in\mathbb{R},$$ $$ -\infty<x,y<\infty.$$ Show ...
1
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0answers
22 views

Getting wrong answer when using the elimination method to find values a and b that make f(x) continuous

We have $$ f(x) = \begin{cases} \dfrac{x^2-9}{x-3}, & x<3\\[6pt] ax^2+bx+21, & 3\leq x <4\\[6pt] x-7, & x\geq 4 \end{cases} $$ We know that $$ ...
0
votes
2answers
17 views

Solution of $y'=xy^{1/3}, y(0)=0$ equal to $0$ in $[-c,c]$ and positive for $|x|>c$.

I'm looking for a continuous function $y(x)$ which satisfies the above and trying to make it depend on $c$ so that a solution exists for any $c>0$. I read it is possible, but I can't do it... Can ...
0
votes
1answer
25 views

Does it exist a function that is continuous at every rational point and discontinuous at every irrational point?And vice versa?

Actually there are 2 questions, but they are closely related. Does it exist a function that is: 1. Continuous at every rational point and discontinuous at every irrational point? 2. Continuous at ...
-1
votes
0answers
24 views

If f(x) is continuous on Interval then f(x) is continuous on (-inf,inf). [closed]

Prove that if $f(x)$ is continuous on $(-a,a)$ for all $a>0$, then $f(x)$ is continuous on $(-\infty ,\infty)$. Prove that if $f(x)$ is uniformly continuous on $(-a, a)$ for all $a>0$, then ...
1
vote
1answer
9 views

Continuity Property Proof Check

Suppose $f$ is continuous at $x_0$ and $f(x_0)>M$. I claim that $f(x)>M$ for all $x$ in some neighborhood of $x_0$. Let $M=f(x_1)$. We have that $\lim_{x\rightarrow x_0}f(x) > f(x_1)$. ...
2
votes
0answers
47 views

A continuous function cannot take every value an exact even number of times?

I have proved that a continuous function $f(x): \mathbb{R} \rightarrow\mathbb{R}$ cannot take every value in its range exactly twice. How is the general case with an even number of times proved? ...
1
vote
2answers
21 views

Proof that a function constant on 2 disjoint closed subsets of $\Bbb R$ with standard topology can be extended to continuos function on $\Bbb R$.

I want to prove that function constant on 2 disjoint closed subsets of $\mathbb{R}$ with standard topology can be extended to continuos function on $\mathbb{R}$. My first attempt was to connect ...
0
votes
0answers
7 views

How to show that this function $H$ is Lipschitz continuous with constant 1

In my book the following statement is given: $H(x)=\mathbb{E}_{\omega}[(\omega-x)^-]$, $x \in \mathbb{R}$ and $\omega$ a one-dimensional random variable and $H$ finite. $H$ is Lipschitz continuous ...
1
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2answers
23 views

Prove continuity using reverse triangle equality

Given $$f(x) = \|x-a\|$$ prove using reverse triangle equality that this is a continuous function. So I proceed like this; we look at the equality $$| f(x) - f(b)|$$ and want to show that it's ...
0
votes
1answer
19 views

Continuity and Differentiability of a series of functions

Consider the function $f(x)=\sum_{n=1}^{\infty} 2^{-n}g(2^{2^{n}}x)$ where \begin{equation} g(x)=\begin{cases} 1+x &-2 \le x \le 0 \\ 1-x &0 \le x \le 2 \end{cases} \end{equation} where ...
0
votes
1answer
12 views

Continuous function and constant sign

Let $f:(a,b)\to\mathbb{R}$ be a continuous function, and $t_0\in (a,b)$ such that $f(t_0)=0$. Is it true that one can find in any case some $\epsilon >0$ such that $f$ has constant sign on ...
0
votes
1answer
31 views

Can we prove the Riemann-Lebesgue lemma by using the Weierstrass approximation theorem?

I'd like to prove the following version of the Riemann-Lebesgue lemma: Let $f: [0,1] \to \mathbb R$ be continuous. Then $$\int_0^1 f(x)\sin(nx) \, dx \xrightarrow{n \to \infty} 0$$ It's quite ...
1
vote
1answer
28 views

Prove f(x)=glb{|x-a| : a in A} is continuous

Let $A \subset R$, let $f(x)=glb{ |x-a| : a \in A}$ -Prove $f$ is well defined -Prove $f$ is continuous (Ok, here's the deal, because of the absolute value the greatest lower bound is always going ...
2
votes
1answer
48 views

A linear operator is continuous if and only if it maps cauchy sequences to cauchy sequences

Let $A$ and $B$ be seminormed spaces, then I want to show that a linear operator $T: A \rightarrow B$ is continuous if and only if it maps cauchy sequences to cauchy sequences. The direction "$T$ ...
1
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0answers
19 views

Is local compactness preserved by continuous closed onto functions? [duplicate]

I've just shown for a homework problem that if $f$ is an open continuous function from $X$ onto a $T_2$-space $Y$, and $X$ is locally compact, then $Y$ is locally compact. I wonder, does this hold for ...
3
votes
2answers
295 views

Is continuity in topology well-defined?

In topology, a function is continuous if inverse of every open set is open. But for the inverse to be well-defined the function should be bijective. For example consider the projection map. It is not ...
1
vote
2answers
37 views

Is a continuous function vanishing at infinity always C_0?

Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be a continuous function with $$ \lim_{|x| \to \infty} f(x) = 0. $$ Does that imply $f \in C_0$, i.e. is there a compact set $K_{\epsilon}$ for every ...
0
votes
1answer
21 views

Using unbounded derivative to show function is not uniformly convergent

I'm confused how to use the following theorem: 19.6 Theorem. Let $f$ be a continuous function on an interval $I$ [$I$ may be bounded or unbounded]. Let $I^◦$ be the interval obtained by removing ...
0
votes
1answer
24 views

Confusion with a proof about the continuity of convex functions

I studying convex analysis and in my book I have the following statement and proof: Lets assume that $f:S\rightarrow \mathbb{R}, \;S\subset \mathbb{R}^n$ is a convex function. Then $f$ is ...
0
votes
2answers
39 views

If $f$ is continuous at $x_0$ and $f(x_0)>M$, then $f(x)>M$ in some neighborhood of $x_0$

If $f$ is continuous at $x_0$ and $f(x_0)>M$, then $f(x)>M$ for all $x$ some neighborhood of $x_0$. My attempt is below. From the assumptions above, we have that $f(x_0) > M = f(x_1)$ for ...
0
votes
3answers
33 views

how to find the smallest s to make f continuous at (0,0)

$$ f(x,y)=\left\{ \begin{array}{lll} \frac{|x|^s|y|^{2s}}{x^2+y^2} & \text{if}& (x,y) \neq (0,0)\\ 0 & \text{otherwise} \end{array} \right. $$ what is the smallest s to make f(x,y) ...
0
votes
1answer
23 views

Find $f$ such that the contraction $\phi$ has a fixed-point $\rho= \sqrt{2}$

I use the Newton method and the Banach fixed-point theorem and have: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous and $f: I \rightarrow \mathbb{R}$ a ...
0
votes
1answer
32 views

Continuity of function proof

Let $f:X \to Y \times Z$ be given by $f(x)=\bigl( f_{1}(x), f_{2}(x) \bigr)$. Prove that $f$ is continuous iff $f_{1}$ and $f_{2}$ are continuous. I'm struggling to relate the pre image of $h$ ...
0
votes
0answers
20 views

Banach fixed-point theorem and Newton

I have to combine the Newton method and the Banach fixed-point theorem: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. Let $f: I \rightarrow ...
0
votes
2answers
60 views

The product of uniformly continuous functions is not necessarily uniformly continuous

I was asked to show that given two functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ which are both uniformly continuous, to show that the product ...
0
votes
4answers
45 views

Limit calculation and discontinuity

Having a function, which has a polynomial in the denominator like: $$ \lim_{x \to 2}\,\dfrac{x+3}{x-2} $$ We see there is a discontinuity at x=2, because it sets the denominator to 0. But ...
1
vote
2answers
51 views

Continuity of a mapping $C\to C^2$, $C$ being the Cantor set

I will denote the Cantor set as $C$. We have proved earlier that every $x\in C$ can be uniquely written in a ternary representation $x=0.a_1a_2a_3...$ where all the $a_i \in \{0,2\}$. Now we consider ...
1
vote
0answers
23 views

Newton method and the Banach fixed-point theorem

I try to combine the Newton method and the Banach fixed-point theorem but I have still some questions: Let $I \subset \mathbb{R}$ a closed interval and $\phi: I \rightarrow I$ Lipschitz continous. ...
0
votes
1answer
27 views

Calculus: Continuous/Differentiable

I have no idea on how to do this problem. $$f(x)= \begin{cases} 2x^2-3x+1& \text{x<1}\\ (x-1)^{\frac{3}{2}}& \text{x $\geqslant$ 1} \end{cases}$$ a. Show that $f$ is continuous at $1$. ...
1
vote
2answers
71 views

Two continuous functions with connected images

Suppose we have two continuous functions $f(x)$ and $g(x)$. Define $f$ on $[0,1]$ and $g$ on $[1,2]$, such that $f(1)=g(1)$. If we know that $\text {Im} (f(x))$ and $\text{Im} (g(x))$ are connected, ...
1
vote
3answers
57 views

Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$

Suppose that $f: \mathbb R \to\mathbb R$ satisfies $f(x + y) = f(x) + f(y)$ for each real $x,y$. Prove $f$ is continuous at $0$ if and only if $f$ is continuous on $\mathbb R$. Proof: ...
3
votes
1answer
63 views

Is there a valid multiplication for any choice of identity in $C(\mathbb{R})$?

Let $C(\mathbb{R})$ be the ring of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Its identity with the usual multiplication is $1(x) = 1$. I have two related questions. Firstly, when we ...
0
votes
1answer
24 views

Necessity in Arzela-Ascoli theorem

I am trying to prove necessity of boundedness and equicontinuity in Arzela-Ascoli and I don't know how to go about it. More precisely,I have: Let $K$ be a compact metric space, and $A\subset C^0(K)$ ...
1
vote
1answer
21 views

If f is a real function, continuous at a and f(a) < M, then there is an open interval I contianing a such that f(x) < M for all x in I.

Can someone please help? If f is a real function which is continuous at a ∈ R and if f(a) < M for some M ∈ R, prove that there is an open interval I containing a such that f(x) < M for all x ∈ ...
0
votes
1answer
15 views

Define $f(y)=d(x_0,y)$, prove that $f$ is continuous.

Consider a metric space $(X,d)$ and some $x_o \in X$. Define function $f_{x_0}(y)=d(x_0,y), $ which is in $\text{R}$. Show that the function is continuous. Here's what I've tried: According to ...
1
vote
3answers
38 views

Is set of all contiuous functions subspace?

This is one of the problems from the book: Hoffman and Kunze, chapter: Vector Spaces Let V be the (real) vector space of all functions f from R into R. Is the set of all f which are continuous, ...