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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Did I miss anything on this question about continuity on the value of $\alpha$

For what value of $\alpha$ is $f(x)$ differentiable at $x = 1$ For each of those values of $\alpha$, find $f^{\prime}(1)$ $$\mbox{Let } f(x) = \begin{cases} 2x-4\tan (x) & x \leq 0 \\ ...
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51 views

A sufficient condition to ensure a function to be linear

Suppose that $f$ is continuously differentiable on $\Bbb R$, and $$\lim_{x\to +\infty}f'(x)$$ exists and is finite. Furthermore, $$f(x+1)-f(x)=f'(x),\ \forall\ x\in\Bbb R.$$ Show that $f$ is linear, ...
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359 views

If $|f(x)-f(y)|\geq \frac12|x-y|$, must $f$ be bijective? [duplicate]

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $$|f(x)-f(y)|\geq \frac12|x-y|$$ for all$x,y\in \mathbb R$. Then is $f$ one-one and onto? Let $f(x)=f(y)$ i.e. ...
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25 views

right inverse and supplement of kernel in a banach

For $T \in L(E,F)$ continuous surjective linear operator between Banach spaces $E$ and $F$ we have that : $Ker(T) $ admits a closed complement $L$ in $E \implies T$ admits a continuous right ...
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85 views

Show that the image of a complete metric space under a continuous map is also complete given an additional condition.

This is a problem from revision material for a functional analysis class. Let $(X,d)$ and $(C,p)$ be two metric spaces and let $f:X\rightarrow C$ be a continuous function with $f(X)=C$. Assuming ...
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3answers
81 views

Continuity of the function defined by: $f(x)=e^x$ if $x$ is rational; $f(x)=e^{1-x}$ if $x$ is irrational

Let the function $f(x)$ be defined as $$f(x)= \begin{cases} e^x & x\text{ is rational} \\ e^{(1-x)} & x\text{ is irrational} \end{cases} $$ for $x$ in $(0,1)$. Then a. $f$ is ...
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1answer
86 views

Analysis-Baby Rudin's differentiability and continuity: theorem 5.2 and 5.6

I am very confused about differentiability and continuity. At the beginning of the differentiation chapter, we proved that differentiability contains continuity. (Theorem 5.2) But in example 5.6 and ...
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32 views

Show that the formula defines an inner product on X

Let $X=C[-1,1]$ be the space of continuous functions $f:[-1,1]\rightarrow \mathbb{R}$. For $f,g\in X$ define: $$\langle f,g\rangle_2=\int_{-1}^{1}|t|f(t)g(t)dt$$ The property i'm struggling with is ...
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4answers
76 views

$\delta$ and $\epsilon$ in the continuity definition

Considering the following definition of continuity (there is nothing unusual yet here): $$\forall \varepsilon > 0\ \exists \delta > 0\ \text{s.t. } 0 < |x - x_0| < \delta \implies |f(x) - ...
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41 views

Is there any noncontinuous function f(x), such that the absolute value of f(x) is continuous? [duplicate]

I am trying to find such a function or a proof, which shows that there is no such function in general. I know, that the other direction of this statement is true. (I prooved it using only the ...
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0answers
64 views

Continuity with $\epsilon$, $\delta$?

My question is the following: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function: \begin{equation} f(x)= \left\{ \begin{array}{ll} {x^2} & \mbox{if } {x \text{ is rational}} \\ {x+2} ...
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1answer
32 views

An zero upper-sum on a function with a discontinuous function

For a Reimann integrabl function $f(x) \geq 0$ on $[a,b]$, I'm asked to... Prove that if $\int_a^b f(x)~dx = 0$ and $f$ is continuous, then $f(x) = 0$ for all $x$ in $[a,b]$. and, immediately ...
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1answer
34 views

Oscillation of function and continuity clarification of proof

There is a similar question related to this, but it doesn't answer my question, so I would be thankful if anyone helped me with it. There is a step specifically in the proof I do not understand. ...
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1answer
29 views

Need help to prove this function

Suppose that $f:R\to$ $R$ is continuous on $R$, and that $f(r)=0$ for every rational number r. Prove that $f(x)=0$ for all $x$ in $R$. I let r in $R-Q$, and $f(x)= lim(f(r_n))=f(lim(r_n))=lim(0)=0$. ...
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0answers
36 views

There is a bijection between $\{ f: X \to Y \text{ continuous} \}$ and $ \tau_{_X} \times \tau _{_Y}$

Let $\tau _{_X}$ and $\tau _{_Y}$ denote the topologies on $X$ and $Y$ respectively. I know that the statement in the topic is not true, but my feelings say that with a small modification, this will ...
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2answers
137 views

Translation operator and continuity

I came across a text that proves that translation operator $T_a(f):=f(x-a)$ where $a\in\mathbb{R}^n$ and $f\in L^p(\mathbb{R}^n)$ is continuous. The proof follows: ...
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1answer
43 views

Extending a function continuously

Let $f:\mathbb{R}\to \mathbb{R}$ be continuous in an interval around some real $c\in (a,b)$. Construct $g:\Bbb{Q}\setminus \{0\}\to \mathbb{R}$ as : $$g(s):=\frac{f(c+s)-f(c)}{s}\quad s\in ...
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25 views

About continuous functions and continuous continuations and their uniqueness

How would you access the following problem: (a) Show that for every $s \in \mathbb{Q}$ the function $$f: \mathbb{C}^* \rightarrow \mathbb{C}$$ $$ f(z) := \frac{\overline z}{\vert z \vert ^s}$$ is ...
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1answer
45 views

Finding an unbounded set with a specific property

Find an unbounded subset $A ⊂ \Bbb R$ such that every function from $A$ to a metric space is uniformly continuous. My attempt at the solution (incomplete). If $A⊆ \Bbb R$ were such a set, then for ...
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If $f$ is continuous on $[0,\infty)$ and not bounded above implies…

$Conj:$ If $f$ is continuous on $[0,\infty)$ and not bounded above implies there exists $\{ x_n\}$ such that $x_n \rightarrow \infty$ and $f(x_n)\rightarrow \infty$. I can see how to show $f(x_n) ...
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54 views

An Application of Intermediate Value Theorem

Let $f :\Bbb R→ \Bbb R$ be given by $f(x) := x^{n}$ for some $n ∈ \Bbb N$. If $b$ is a positive real number, show that there exists a unique positive real number $a$ such that $a^{n} = b$. My ...
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How to show that $F(x)$ continuous?

$F:\Big(C[0,1],||.||_2\Big)\rightarrow \Big(C[0,1],||.||_3\Big)$ $x\rightarrow F(x)(t)=\int^t_0x(s)ds,\quad\quad0 \le t\le 1 $ Show that F is continuous. F is linear. for n=0,1,2.. $x_n(t)=t^n,0 ...
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1answer
64 views

Real analysis. Uniformly continuous

Suppose $$f:\mathbb R\to\mathbb C$$ is continuous and $f(x)=0$ for all $|x|>1$. Show $f$ is uniformly continuous on $\mathbb R$. This is not homework. I'm trying to study for a test. I ...
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2answers
83 views

Infinitely many times differentiable function with unbounded derivatives?

Let $f$ be an infinitely many times continuously differentiable function on the compact interval $[0,1]$. We denote by $f^{(k)}$ the $k$-th derivative with respect to $x$. Then we know: $\sup_{x \in ...
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1answer
47 views

An obstacle in a proof of Lagrange's mean value theorem by Nested Interval theorem

I was trying to prove Lagrange's mean value theorem by Nested Interval theorem and there's step where I got stuck ; let me write down to the step Let $f:[x_1,x_2]\to \mathbb R$ be continuous on ...
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1answer
38 views

Proving uniform continuity using limits

Hi I am interested in a result which states that if a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ has finite limits on both sides then the function is uniformly continuous. Is it ...
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65 views

Function for which it is unknown whether it is continuous

Is there any function $f:\mathbb R\rightarrow \mathbb R$ for which at least some values are known but it is unknown whether $f$ is continuous or not? Edit: I am looking for examples from actual ...
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2answers
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Why does this specific $\delta$ imply a failure of $|f(x)-f(x_0)|< \epsilon$? (trouble in understanding continuity proof)

I'm having a bit of trouble with a the proof in Ross' Elementary Analysis. The theorem is the $\epsilon-\delta$ one. Theorem: Let $f$ be continuous at $x_0$ in $dom(f)$ if and only if for each ...
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Every function from discrete metric space to another metric space is uniformly continuous

My solution:It is fairly straightforward graphically but I just want to ensure if it is rigorous enough. Suppose $X$ is a discrete metric space and $f$ be any function from $X$ to $Y$ where $Y$ is ...
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1answer
46 views

proving C2 continuity given closed spline constraints

Given the closed spline's constraints as below $$P(0) = P_k $$ $$P(1) = P_{k+1}$$ $$P''(0) = P_{k-1} - 2P_{k}+P_{k+1}$$ $$P''(1) = P_{k} - 2P_{k+1}+P_{k+2}$$ How do I prove that this spline ...
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31 views

Question regarding the sequence definition of continuity.

Here is an excerpt from Ross' Elementary Analysis (specifically the definition of continuity): "The function $f$ is $\it \space continuous\space at \space x_0$ if, for every sequence $(x_n)$ in ...
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A closed, bounded subset $A$ of $\Bbb Q$ and a continuous function $f : A → \Bbb R$ such that $f$ is not bounded

Find a closed, bounded subset $A$ of $\Bbb Q$ and a continuous function $f : A →\Bbb R$ such that $f$ is not bounded Note: $\Bbb Q$ is the set of all rationals. My Solution: $A=\{x:x\in\Bbb Q, ...
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74 views

A continuous path between shapes

Let $A$ and $B$ be two geometric shapes in the plane (two measureable subsets of $\mathbb{R}^2$) such that $A\subseteq B$. Define a $path$ from $A$ to $B$ as a function $f$ from $[0,1]$ to subsets of ...
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1answer
32 views

Continuity of $\sup_{x\in\Omega}\varphi(x,\cdot)$

Let $\Omega\subset\mathbb{R}^n$ be open,bounded and (I don't know if this matter) of class $C^{1+\alpha}$. Let $\varphi:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $\varphi(x,\cdot)$ is ...
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1answer
469 views

Prove sin(1/x) is discontinuous at 0 using epsilon delta definition of continuity

Let $$f(x) = \begin{cases} 0 &\text{ if $x=0$}\\ \sin(1/x) &\text{ otherwise} \end{cases} $$ Prove that $f$ is discontinuous at $0$ using the $\epsilon \delta$ definition of continuity. I ...
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2answers
58 views

Lipschitz continuity of a function

Show that $g(x) = \frac{1}{x^{2} +1}$ is Lipschitz conitnuous. From the definition, we must show that $\forall x,y \in \mathbb{R}$, $|f(x)-f(y)| \leq K|x-y|$, for some real constant $K$. First, I ...
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1answer
107 views

If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .

Exercise: Suppose that $f$ is continuous on $[a,b]$ and that $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ . Attempt of proof: Suppose that $f$ is continuous on $[a,b]$ and that ...
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1answer
27 views

Show that the function is discontinuous at c=2 using delta epsilon

I am struggling with discontinuity and continuity. f(x)= {-1, x<2 0, x=2 1, x>2 I realize that I must show that $\exists$ $\epsilon$ > 0 ...
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Prove continuity of $\frac{xy}{x^2 + y^2}$ using formal definition

I need to prove that the function $f(x, y) = \frac{xy}{x^2 + y^2}$ is continuous on $(x,y) \in \mathbb{R}^2 - (0,0)$ using the following definition of continuity: Let S be a subset of $\mathbb{R}^p$. ...
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2answers
63 views

How do we 'know' that $2^x$ is continuous?

It is intuitive for $2^n$, if $n$ is an integer, to exist. How do we know that less intuitive values such as $2^\frac{1}{2}$, $2^\sqrt{2}$, $2^\pi$ etc exist? I'd like to accept that $2^x$ is ...
2
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1answer
50 views

Continuity of $e^{-x^2/4k}$

I want to show that $f(x)=e^{-x^2/4k}$ (where $k>0$ is fixed) is continuous using an $\epsilon$, $\delta$ argument. I've been trying to choose $\delta$ using $\ln$ somehow and I've also been trying ...
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1answer
42 views

Continuity from property of constriction images of spheres

Let $D\subset\mathbb R^n$ --- domain and mapping $\varphi:D\to \mathbb R^n$. The following property holds There is a set $T\subset D$ s. t. measure $|D\setminus T|=0$ and for every point ...
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1answer
28 views

Mulltivariable limit quesion and my attempt

Given $$ F(x,y) = \left\{ \begin{array}{ll} 0 & (x,y)=(2y,y) \\ \exp \biggl( \frac{|x-2y|}{x^2 -4xy +4y^2} \biggr) & (x, y) \ne (2y, y) \end{array} \right. $$ Task is to examine whether ...
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1answer
109 views

Continuity and differentiability of two variables function

Let be $f:\mathbb{R^2}\rightarrow\mathbb{R}$ defined by: $$f(x,y)= \begin{cases} x^3\log{\left(1+\frac{|y|^\alpha}{x^4}\right)} & \text{if } x \neq 0 \\ 0 & \text{if } x =0 \end{cases}$$ ...
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0answers
86 views

Extending a continuous function to the closure

I'm dealing with the following problem: Let $X$ a topological space, $Y$ a metric space and $A$ a subspace of $X$. If $f$ is a continuous mapping of $A$ into $Y$, show that $f$ can be extended in ...
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3answers
65 views

trying to prove: If $f$ is continious and is lebesgue-almost-everywhere constant, then it is constant

I was wondering if this claim is true, and if it is then how would one try to prove it: If $f\in C[0,1]$ (and thus is continuous) and is Lebesgue-almost-everywhere constant, then it is constant. It ...
0
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1answer
20 views

semi continious functions characterizations

Does anyone knows how to prove this: Let $f: (X, d) \rightarrow \mathbb{R}$ be an upper semi-continious function. Prove that $f$ is u.s.c. if and only if $ \{ x \ \ |\ \ f(x) \geq z \} $ is closed ...
0
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1answer
29 views

Show there is an unbounded continuous function on a closed nonempty subset of a metric space.

Let $X$ be a metric space. Let $E \subset X$ be not closed and nonempty. Show that there is a continuous real-valued function on $E$ that is not bounded. The only function that I know of that is ...
2
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1answer
27 views

Continuity in argument of minimization

Let $$g(c) = \min_{Ax=c} f(x),$$ where $x$, $c$ are vector-valued, $A$ is a matrix and $f$ is a smooth convex function. Under what conditions can we say $g(c)$ is continuous in $c$?
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1answer
37 views

Principal bundle map is fiber homeomorphism

let $B_1(\mathcal{P}_1:P_1\rightarrow X_1)$ and $B_2$ be two principal G-bundles and let $\tilde f:P_1 \rightarrow P_2$ be a principal bundle map. I want to prove that $\tilde f$ carries each fiber of ...