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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16 views

Analytic continuation of function continuous on boundary

Suppose one has a function $f$ in the disc algebra ie: $f$ is continuous on $|z|\leq1$ and holomorphic in $|z|<1$. I wondered, can $f$ always be extended to a holomorphic function on some region ...
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16 views

Bounded Derivatives and Uniformly Continuous Functions

Prove or Disprove: Let $f:\mathbb{R} \to \mathbb{R}$ be a bounded uniformly continuous function that whose first and second derivative exists and is continuous, in other words $f \in C^2_{unif} ...
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3answers
48 views

Let $A = \{\frac{1}{n}:n\in\mathbb{N}\}$. Prove that $f:A\to \mathbb{R}$ is continuous.

Let $A = \{\frac{1}{n}:n\in\mathbb{N}\}$. Suppose $f:A\to \mathbb{R}$. Prove $f$ is continuous on $A$. Definition of continuity: for all $\varepsilon>0$,there exists a $\delta>0$ such that ...
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1answer
58 views

Meaning of $\gamma=0+$ notation

Here $v$ is continuous at $(0,\rho)$ and $v$ is smooth $\forall \rho$, $\gamma \neq 0$ what is the meaning of $\gamma=0+$ in the following context: If $v(\gamma=0+, \rho)=v(\gamma=0-,\rho)$ ...
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1answer
26 views

Continuity in $\mathbb{R^2}$ notation

If $u(\xi=0+, \eta)=u(\xi=0-,\eta)$ Does this mean $\lim \limits_{\xi \to 0+}u(\xi,\eta)=\lim \limits_{\xi \to 0-}u(\xi,\eta)$ ?
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63 views

Separation in compact spaces

There was recently a question that I cannot find about separation in compact spaces. The answer to that question was no for trivial reasons. Motivated by that, let me ask a less trivial version of ...
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0answers
10 views

Continuity of a solution to a pde

If $u(x,t)=1$ for $x>t$ and $u(x,t)=\frac{1}{1+\tau^2}$ for $x<t$ with $\tau=\frac{\sqrt{1+4x(t-x)}-1}{2x}$ how can I check that $u$ is continous at $x=t$? If I sub in $x=t$ for ...
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2answers
23 views

Show that the given function is a uniformly continuous function.

Let $F : \mathbb{R}^{n} → \mathbb{R}$ be defined by $F(x_1, x_2, . . . , x_n) = \max\{|x_1|, |x_2|, . . . , |x_n|\}$. Show that $F$ is a uniformly continuous function. I really have nothing to show ...
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3answers
33 views

Manipulating inequalities in epsilon delta

I need to show that the polynomial $$x^3-x-3$$ is continuous at $x=1$ using epsilon delta proof but I'm facing some problem manipulating the inequality. Given $$\epsilon>0$$ ...
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0answers
24 views

sequential continuity and countinuity

When we have two topological spaces, $\left(X, \tau_X\right)$ and $(Y, \tau_Y)$ it is easy to check that for $f: X \rightarrow Y$ continuity implies sequential continuity. I'm wondering what do we ...
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2answers
43 views

Proving that $\lim \limits_{x \to a^+} f(x)$ exists

We have $f : (a,b) \to \mathbb{R}$ with the following property $$|f(x) - f(y)| \leq M |x - y|^{1/2}$$ for $x, y \in (a,b)$ and a constant $M$. Prove that $\lim \limits_{x \to a^+} f(x)$ exists. ...
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1answer
20 views

Find the constants so the function is continuous [on hold]

I'm having a little bit of trouble on this problem: Find the constants a, b, c so that f(x) is continuous on x = -1 f(x) = { x² + ax + b / x + 1 if x < -1 { c if x=-1 { x² + 5x if x > -1 Can ...
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2answers
31 views

extension of a continuous function

Suppose $f:X\to Y$ is a continuous map between two metric spaces. Can we extend $f$ to a function $f':X'\to Y'$ in such a way that $f'$ is also continuous ($X'$ and $Y'$ are also metric spaces), where ...
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1answer
24 views

Subalgebra of $C(X)$ that separates the points

I try to prove that Let $X$ be a compact space and $C(X) = \{f \colon X \to \mathbb F \mid f$ is continuous$\}$. Suppose that $ \mathscr A $ is a subalgebra of $ C(X) $ that separates the points ...
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2answers
18 views

Show under these conditions that $f$ is uniformly continuous

Let $f: ]a, b[ \to \mathbb{R}$ be differentiable and let there be an $M > 0$ such that $|f'| \leq M$ on $]a, b[.$ Then $f$ is uniformly continuous on $]a, b[.$ By differentiability, the function ...
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0answers
15 views

Proof involving strong continuous semigroups

Let $T(t)$ be a $C_{0}$ semigroup on the Hilbert space $X$ with infinitesimal generator $A$ and let $\rho\in(0,1)$. I want to prove that $\displaystyle \sup_{t\ge 0}||T(t)-I||\le \rho$ is equivalent ...
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14 views

separation in perfectly normal spaces [on hold]

Suppose that we have two distinct points $x$ and $y$ in a perfectly normal compact space. Can we find a norm one continuous function such that $f$ equals to $1$ in some neighborhood of $x$, $f$ equals ...
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2answers
82 views

How do we determine if $f '(0)$ exists [duplicate]

Suppose that f: $\mathbb{R} \to \mathbb{R}$ is continuous and $f '(x)$ exists $\forall x \gt 0$ and $\lim_{x\to 0} f '(x) = 3$. Does $f '(0)$ exist? So it's apparent that my function $f$ is ...
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0answers
15 views

Prove that the given subset satisfying the given hypothesis is compact.

Let C be a subset of a compact metric space (X, d). Assume that, for every continuous function h : X → R, the restriction of h to C attains a maximum on C. Prove that C is compact. My attempt: I ...
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1answer
12 views

Discontinuities and their locations

So we were given a problem that states Let $\lfloor x\rfloor$ be the greatest integer $\leq x and let (x) = x-\lfloor x\rfloor$ be the fractional part of x. Identify the location and type of ...
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1answer
24 views

Let $f : R → R$. Sequence in image converges, prove convergence of the function at a given value in the sequence.

Let $f : R → R$. Assume $f$ is increasing. Assume $f(1) = 2$. Assume the sequence $2 + (−1)^n/n$ belongs to the image of $f$. Prove that $f$ is continuous at $1$. Should I just show the sequence ...
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14 views

True/False with regard to uniform continuity on sum of

I am having a lot of trouble proving or disproving these. Let $f_n : E → \Bbb R$ be continuous functions for $1 ≤ n ≤ N$. Let $a_k^{ (n)}$ be $N$ convergent sequences of numbers. Assume $lim_{k→∞} ...
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2answers
15 views

Finding the infinitesimal generation of a strongly continuous semigroup

Let $X$ be a Hilbert space, $A\in\mathcal{L}(X)$ and $\displaystyle T(t)=e^{At}=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}$. I have already shown that $T(t)$ defines a $C_{0}$ semigroup. But now I need ...
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1answer
60 views

Showing a function can not be continuous.

I want to show that there does not exist a continuous function $f(x)$ satisfying the following criteria. $$ \int_0^{1/2} f(x) dx - \int_{1/2}^1 f(x) dx = 1 $$ When we restrict $\displaystyle \|f(x)\| ...
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2answers
53 views

Continuity of this function at a point

let $f(x)=x^2$ defined on $[0,1]$ my question may seem silly but i am really confused about it. if we want to prove the continuity of $f$ at $x=1$ then we should have: $\lim f(x)$ as x tends to ...
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1answer
36 views

Why is $A:X\to Y$ linear between two normed spaces is continuous iff bounded?

Why is it that every linear operator $f: \mathbb R^n \to \mathbb R^m$ is bounded and therefore continuous, but why is it that $A:X \to Y$ between two normed spaces is continuous iff bounded? That ...
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1answer
60 views

How to visualize the limit of this function?

$$ f(x) = \begin{cases} x, & \text{$x$ rational} \\ -x, & \text{$x$ irrational} \end{cases} $$ $ \text{This function is not continuous at any point except 0.} $ Intuitively, I am able to ...
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41 views

Discontinuous function counter example

When providing counter-examples for various things in Calculus, we often utilise piecemeal functions because we can easily 'construct' something 'pathological' by doing that. Somebody asked me "To ...
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15 views

Showing that an operator generates a unitary group

Consider the following operator on $X=L^{2}(0,1)$: $\displaystyle Af=\frac{df}{d\zeta}$ with domain: $D(A)=\{f\in L^{2}(0,1)|f$ is absolutely continuous, $\frac{df}{d\zeta}\in L^{2}(0,1)$ and ...
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2answers
58 views

Is there a non-decreasing function that is discontinuous at every rational point? [duplicate]

A well-known theorem is that if $f:[a,b]\to\mathbb{R}$ is non-decreasing, then $f$ as at most countably many discontinuities. This led me think of the following question. Question: Is there a ...
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38 views

Continuity and interior

I have questions about the relation between continuity and interior based on the article ;Continuity and Closure At first I guess that there will be a property like $f:X\rightarrow Y$ is continuous ...
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26 views

Is a function induced by an absolutely continuous measure on $\mathbb{R}^d$ continuous

Let $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d),\mathbb{P}_X)$ be a probability space, where $\mathbb{P}_X$ is absolutely continuous w.r.t. Lebesgue measure $\lambda$ on $\mathbb{R}^d$, i.e. ...
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1answer
14 views

Continuity of a composition of continuous functions

Suppose $f: \mathbb{R} \to \mathbb{R} $ is continuous at $x = 1 $ and $g: \mathbb{R} \to \mathbb{R} $ continuous at $y = f(1) $. Then $g \circ f $ is continuous at $x = 1 $ Attempt: Let $\epsilon ...
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1answer
11 views

Proving one of the properties of a strongly continuous semigroup

Let $X$ be a Hilbert space and $A\in\mathcal{L}(X)$ and $\displaystyle T(t)=e^{At}=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}$. I want to show that $T(t+\tau)=T(t)T(\tau)$. So we have that ...
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39 views

How to show that in the category of Hausdorff spaces every epimorphism is a continuous function with dense image? [duplicate]

How to show that in the category of Hausdorff spaces every epimorphism is a continuous function with dense image ? that is if $X,Y$ are Hausdorff spaces and $f:X \to Y$ is continuous such that for any ...
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1answer
30 views

Continuous function on interval $[0, \infty]$

Given function $f :[0, \infty] \rightarrow \mathbb{R}$. We know that $f$ is uniformly continuous on interval $(0, \infty]$ and continuous on point $0$. How to prove that $f$ is uniformly continuous ...
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56 views

What does it mean to make a function continuous?

Can we make $\frac{\sin (x+y)}{x+y}$ continuous, defining it appropriately at $(0, 0)$ ?? What does it mean to make a function continuous??
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27 views

Showing Function is Continuous

Let $f: \mathbb{R} \backslash \{2\} \to \mathbb{R}$ be the function given by $f(x) = \frac{2x^2+x-10}{3x-6}$. Let $g: \mathbb{R} \to \mathbb{R}$ given by: $$g(x)=\begin{cases}{f(x)} & \text{if } ...
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1answer
79 views

showing that $w$ is continuous at $1$

Let $w\colon\mathbb{R}\to\mathbb{R}$. Assume $w$ is increasing. Assume $w(1) = 2$. Assume the sequence $2 + \frac{(−1)^n}{n}$ belongs to the image of $w$. How is $w$ continuous at $1$? This doesn't ...
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1answer
17 views

Has the functions having countably infinite image, but finite when the domain is bounded, a conventional name?

I'm trying to find properties for functions that cover the following properties and wondering if they have a formal name to search more efficiently. The function $f(x)$ cover the following ...
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52 views

Why is an automorphism of $\mathbb R$ continuous

I was trying to understand this answer here but got stuck. It's clear to me that $\varphi: \mathbb R \to \mathbb R$ should map positive numbers to positive numbers and that it follows from that that ...
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1answer
32 views

Discontinuous function proof using $\epsilon - \delta$ [duplicate]

$f:\Bbb R \to \Bbb R$ Showing $f(x)=\left\{\begin{array}{cc} 3x,&x\in\Bbb Q\\-3x,&x\in \Bbb I\end{array}\right.$ I want to show that $f(x)$ is discontinuous for all $x\ne0$ using $\epsilon ...
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1answer
32 views

How can we apply the definition?

Show that $$g(x, y)=ye^x+\sin x+(xy)^4$$ is continuous. The definition is: $f : A \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $x_0 \in A$ iff $\forall \epsilon \exists \delta:$ ...
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0answers
38 views

g(x) = 1/(1+x^2) continuous everywhere (is this solution correct) [duplicate]

how would you prove that g(x) = 1/(1+x^2) is continuous everywhere. I have the following: g is continuous at point a if for all ε > 0 there exists 𝛿 > 0 such that for all a in R, |x-a| < 𝛿 then ...
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90 views

$g(x) = 1/(1+x^2)$ is continuous everywhere epsilon delta approach

I have a function g: R→R given by the function $g(x) = 1/(1+x^2)$. I want to prove that this is continuous everywhere. I was reading my real analysis textbook and it seems like a great approach ...
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4answers
86 views

There's no continuous injection from the unit circle to $\mathbb R$

I read a proof that goes as follows: Let $U$ be the unit circle, and let $f : U \longrightarrow \mathbb R$ be a continuous mapping. $U$ is compact and connected, so $f(U)$ is a closed, bounded ...
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242 views

Are absolutely continuous functions piece-wise monotone?

I was wondering whether absolutely continuous functions $f\colon\mathbb{R}\rightarrow \mathbb{R}$ are piecewise monotone. Thanks
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51 views

If $f:[a,b]\rightarrow \mathbb{R}$ is continuous on $[a,b]$ and $f(a)\neq f(b)$, then $f$ is stricly monotonic on some segment $[c,d]\subseteq [a,b]$?

Is the following statement true: If $f:[a,b]\rightarrow \mathbb{R}$ is continuous on $[a,b]$ and $f(a)\neq f(b)$, then $f$ is stricly monotonic on some segment $[c,d]\subseteq [a,b]$? It seems ...
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0answers
20 views

Show that f is not differentiable at the origin of the following function.

Show that f is not differentiable at the origin of the following function: $f(x,y) = \left\{\begin{matrix}\frac{2xy}{x^2+y^2}, (x,y) \neq (0,0)\\ 0, (x,y) = (0,0) \end{matrix}\right.$ I was thinking ...
0
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3answers
67 views

Show $\frac{1}{n^{0.5}}$ is continuous

Show $\frac{1}{n^{0.5}}$ is continuous for $[1,\infty]$. I am unsure how to go about showing this, anyone have any ideas?