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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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
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1answer
38 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 "$T$ ...
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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 ...
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2answers
280 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 ...
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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
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1answer
17 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 ...
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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 ...
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2answers
34 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 ...
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3answers
26 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
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1answer
22 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 ...
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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 $h$ ...
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0answers
17 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 ...
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2answers
54 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
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4answers
43 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 ...
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2answers
50 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 ...
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0answers
20 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
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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 $1$. ...
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2answers
70 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, ...
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56 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: ...
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1answer
36 views

Continuity vs differentiability [on hold]

If a derivative is increasing on a given interval, is it then also continuous on that interval? I.e. $f'(x)$ is increasing on $[a,b]$. Is $f'(x)$ continuous on $[a,b]$?
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0answers
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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 ...
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1answer
22 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 $A\subset C^0(K)$ ...
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1answer
17 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 ∈ ...
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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 ...
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3answers
35 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, ...
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31 views

example of two continuous real-valued functions whose product is 0

Is there an example of two continuous real-valued functions, say on some interval, whose product is 0?
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21 views

prove that $f(x,y) = x^2+y^2$ is continuous on rectangle R.

where $R = \{(x,y): |x|, |y| \leq \frac{1}{\sqrt 2} \}$ I am trying to use picard's theorem so I have to prove that f is continuous on R and that it's lipschitz continuous. How would I do this? I ...
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2answers
32 views

Unbounded function on compact interval?

So what are some unbounded function on compact interval, if there is any? Also, is the function $f:[0,\infty) \to \mathbb R$, $f(x)=x$ continuous?
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1answer
27 views

If $f$ is continuous on a bounded closed interval, then the supremum of $|f|$ is finite

If $f \colon [a,b] \to \mathbb{R}$ is continuous, then $\sup_{x ∈ [a,b]}\left | f(x)\right |$ is finite. Attempt: Suppose $f\colon [a,b] \to \mathbb{R}$ is continuous, then by the Extreme value ...
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2answers
28 views

The plane minus the graph of a continuous function consists of two path-connected components?

Let $f:\Bbb R\rightarrow \Bbb R$ be continuous. Show that $\Bbb R^2-\mathrm{graph}(f)$ consists of two path-connected components. I can show that the area 'above' the graph of $f$ and the area ...
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84 views

Prove that $f(x)$ is a constant function.

Here is the question: Let f be a real valued continuous function on $[0, ∞)$. Suppose $f (x) = f (x^2)$ for all x ≥ 0, prove that f (x) is a constant function. My attempt: Since f(x) is continuous, ...
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1answer
26 views

What are the continuous functions that satisfy the following?

$f(x) = \begin{cases} 0, & x < 0 \\ 1 - f\left(\dfrac{1}{x}\right), & x > 0\text{.} \end{cases}$ I want this to generate a random variable that will be used as a proportion in a way ...
17
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1answer
335 views

A function having limit at every point but continuous nowhere

Is there a function $\,f:\mathbb{R}\rightarrow\mathbb{R},\,$ that has a limit at every point but is continuous nowhere?
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1answer
17 views

Distance to a set

I have a question concerning to the following problem. Let $(X,d)$ be a metric set. For every subset $T \subset X$ we define a mapping \begin{equation} d_T : X \rightarrow R , d_T(x) := inf\{d(x,y) | ...
2
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1answer
17 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
3
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1answer
48 views

Show that $f$ is continuous if it follows the intermediate value property

If $f: [a,b] \to \mathbb{R}$ is $1-1$ and has the intermediate-value property -- that is, if $y$ is between $f(u)$ and $f(v)$, there is at least one $x$ between $u$ and $v$ such that $f(x)=y$ -- show ...
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5answers
147 views

Looking for an example of a bijective continuous function $f:\mathbb{Q} \to \mathbb{Q}$ such that $f(-1)=0$, $f(0)=1$ and $f(1)=-1$?

Clearly such a function does not exist from $\mathbb{R}$ to itself, but apparently it does in $\mathbb{Q}$ and I don't see how it could... Can you give me an example and explain to me how you thought ...
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1answer
14 views

Continuity of function and its value.

Here's a problem I'm struggling with. Not really sure how to do this. My tools are epsilon delta proofs for continuity and that's about it. Let $f:[0,\infty)\to\Bbb R$ be a function which is ...
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2answers
12 views

Bounded - Continuous Relation

How to solve the following question? $$$$ Suppose $f:A\subset\Bbb{R}^2\to\Bbb{R}$ continuous in the rectangle $A=\{(x,y)\in\Bbb{R}^2|\alpha\leq x\leq\beta;\alpha'\leq y\leq\beta'\}.$ Proof that $f$ ...
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108 views

Fundamental limit in two variables

Can I write that $$\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=\lim_{u\to0}\frac{\sin(u)}{u}$$ and, hence, that $\lim_{(x,y)\to(0,0)}\frac{\sin(x^2+y^2)}{x^2+y^2}=1$? If so, why can I do it?
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1answer
26 views

Continuity with restrictions

Suppose that $f \colon A \to \mathbb{R}$ is a function and that $B \subseteq A$. We define the restriction of $f$ to $B$ to be the function $f|_B B \to \mathbb{R}$ defined by $f_B(x) = f(x)$ for all ...
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136 views

Are differentiation and integration continuous functions?

Is differentiation a continuous function from $C^1[a,b] \to C[a,b]$? I think it is but I can't prove it... Would it be possible to prove it using theory about closed sets in $C[a,b]$ and their ...
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2answers
34 views

Spectral Measures: Support vs. Norm

Given a complex Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ and its associated normal operator: $$T:=\int_\mathbb{C}zdE(z)$$ ...
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1answer
19 views

Composition of two continuous functions is continuous

Let $f, g$ be functions $f$ is continuous at a, $\operatorname{f}(a) = b \in \operatorname{Dom}(f)$, $g$ is continuous at b. Then $g\circ f$ is continuous at $a$. Proof: Let $\varepsilon>0$, ...
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76 views

Continuity and Differentiation on open interval

$$f(x) = \begin{cases} x\sin(1/x), & \text{if $x$ $\ne$ $0$} \\ 0, & \text{if $x$ = $0$} \\ \end{cases}$$ Is $f$ continuous on $(-1/\pi$, 1/$\pi$)? Is $f$ differentiable on $(-1/\pi$, ...
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3answers
29 views

the map $f:[0,1]\to [a,b]$ $f(x,y)=(1-x)a+xb$ is a homeomorphism

A question I just came across : A bijection $f:X\to Y$ is a homeomorphism if $f$ and $f^{-1}$ are continuous . Show that the map $f:[0,1]\to [a,b]$ $$f(x)=(1-x)a+xb$$ is a homomorphism... I ...
2
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1answer
22 views

Doubt on understanding continuity .

Just preparing for my multivariable-calculus exam and wanted to clear these things: I've come across many questions of sort below ,especially 2-dimensional regions, and wanted to understand the ...
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2answers
30 views

Prove that a function is continuous in R^2

Prove that $f$ is continuous at $(0,y_0)$ where $f$ is defined on $\mathbb{R}^2$ by $$ \begin{cases} (1+xy)^{1/x} & x\neq 0 \\ e^y & x=0 \\ \end{cases} $$ I'm not really ...
5
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2answers
50 views

Continuity of $f$ on $\mathbb R^2$

The question says: Let $$f(x,y) = \begin{cases} \dfrac{\text{sin}^2(x-y)}{|x|+|y|} & \text{if $|x|+|y|>0$} \\ 0 & \text{if $|x|+|y|=0$} \end{cases}$$ Is $f$ continuous on $\mathbb R^2$? ...
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0answers
11 views

Regarding functions from R² to R: continuity and differentiability

Let $f : U \rightarrow \mathbb{R}$ where $U \subseteq \mathbb{R}^2$ is an open set and $P \in U$. I am almost sure the following statements are correct, but please confirm: The only requirement for ...