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 the function to not be continuous at $z = 0$

$$f(3) = \begin{cases} \dfrac{\mathrm{Re}(z)}{|z|} & \text{when $z \neq 0$} \\ 0 & \text{when $z = 0$} \end{cases}$$ Can someone please explain the concept behind solving such a problem? ...
4
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2answers
109 views

Where surjectivity goes in?

Let $X$ be an infinite set with the cofinite topology, and $f: X \to X$ a surjective function. Prove that $f$ is continuous if and only if $f^{-1}(\{x\})$ is finite for all $x\in X$. I know that $f$ ...
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2answers
57 views

Confusion about Lusin's Theorem.

I saw a proof which heavily relied on Lusin's Theorem recently, and I was hoping someone might be able to help me fill in the detail as to why this theorem allows for a particular creation. ...
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1answer
77 views

Finding a denumerable set $X_0$ satisfying a condition.

Let $(X,\tau)$, with $X$ an uncountable set, $x_0 \in X$ fixed, be the space with topology generated by the collection: $$\mathscr{B} = \{ \{x\} \mid x \in X \setminus \{x_0\}\} \cup \{ A \subset X ...
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2answers
62 views

If $g$ is continuous then $x^ng(x)$ converges on $[0,1)$

Suppose $g:[0,1]\to\mathbb R$ is a continuous function satisfying $g(1)=0$. Prove that the functions $f_n(x)=x^ng(x)$ converge uniformly on $[0,1]$. Hence or using Mean Value Theorem, prove that if ...
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1answer
60 views

Is $\{(x,y) \in \mathbb R^2 : xy=0 \}$ homeomorphic to $\mathbb R$?

Is $\{(x,0) : x \in \mathbb R \} \cup \{(0,y) : y \in \mathbb R \}$ homeomorphic to $\mathbb R$ ? I am totally stuck and I don't even have any intuition whether they should be homeomorphic or not . ...
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2answers
32 views

To find the value of a function at a point where it is continuous

Find $f(0)$ so that the function $f(x)=\dfrac{1-\cos(1-\cos x)}{x^4}$ is continuous everywhere. My attempt: By applying sandwich theorem $-1 \le \cos(x) \le 1$. $$1 \ge -\cos(x) \ge -1$$ $$2 \ge ...
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2answers
21 views

Families of continuous functions with non-continuous derivatives

What families of functions have the property of being continuous yet having a non-continuous derivative? And how many of these families are there? $$f(x) = \sqrt[n]{x}$$ when "n" is an odd number ...
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2answers
31 views

Finding the point at which a function is continuous

I am trying to understand the solution to the following question. At which $c\in\mathbb{R}$ is the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by $$f(x)=\begin{cases}x&\text{if $x$ is ...
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1answer
33 views

Show a series of functions is discontinuous at a point

I have a series of functions which converges to an integrable function. I need to show that this function is discontinuous at every point . For starters (because of the way it's defined) I'm just ...
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1answer
84 views

Monotone increasing continuous function with $\int_a^b f' = f(b) - f(a)$ which is not absolutely continuous

If $f:[a, b] \to \mathbb{R}$ is continuous and real-valued, f' integrable on [a, b], and $\int_a^b f' = f(b) - f(a)$, must f be absolutely continuous? What if f is monotone increasing? For the ...
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1answer
44 views

Relation between Lipschitz condition and linear growth condition

If for a function $f:\mathbb{R}\rightarrow\mathbb{R}$ it is given that it satisfies a Lipschitz condition $\big|f(x)-f(y)\big| \le L\big|x-y\big|$, for all $x,y\in\mathbb{R}$, can we say anything ...
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0answers
50 views

Absolutely continuous iff continuous of bounded variation

I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ...
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0answers
25 views

continuity at a point in a cadlag process

I am reading a proof that uses the fact: Let $(X_t)_{t \geq 0}$ be a cadlag process. We know that $X(\omega)$ has at most countably many discontinuities, for each $\omega \in \Omega$. It is then ...
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4answers
167 views

$f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous , and $f(x+1)+f(x)=x^2$

I would appreciate if somebody could help me with the following problem: Find $f(x)$ ($f(x)$ is not Polynomial function), given that: $f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is ...
1
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2answers
59 views

A connected space that admits a nonconstant continuous map into reals is uncountable

Let $f:X\to Y$ be a non-constant continuous map of topological spaces. If $Y=\mathbb{R}$ and $X$ is connected then $X$ is uncountable. True or False? I know that $f(X)$ is an interval. The ...
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3answers
65 views

If $\operatorname{id}:(X,d_1) \to (X,d_2)$ is continuous for any two metrics $d_1$ and $d_2$, then what will be $X$?

Let $X$ be a set with the property that for any two metrics $d_1$, and $d_2$ on $X$, the identity map $\operatorname{id} : (X, d_1) \to (X, d_2)$ is continuous. Which of the following are true? ...
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1answer
24 views

Send 3 space minus a ring onto the circle

Here's a topology problem I'm having trouble solving. I'm sure it's something simple. Let $S \subset \mathbb{R}^3$ be $\{z=0; x^2 + y^2 =1\}$. Show that there is a continuous function from ...
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2answers
101 views

Is $f(x) = \sum^{\infty}_{n=1} \sqrt{x} e^{-n^2 x}$ continuous?. Where is bluff?

I have a function defined by $f(x) = \sum^{\infty}_{n=1} \sqrt{x} e^{-n^2 x}$. The task is to check, whether $f(x)$ is continuous at $x = 0$. I have proposition of a solution and I would like someone ...
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1answer
45 views

Proving continuity of $f(x,y)$

Let $I=[0,1]$. Let $Q=I\times I$. Define $f:Q \to \mathbb{R}$ by letting $f(x,y)=1/q$ if $y$ is rational and $x=p/q$, where $p$ and $q$ are positive integers with no common factor; let $f(x,y)=0$ ...
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1answer
45 views

Suppose that $f : \mathbb{R}\rightarrow \mathbb{R}$ is continuous and that $f(x)\in \mathbb{Q}$ for all $x \in \mathbb{R}$. Prove $f$ is constant. [duplicate]

I am really stuck on this proving this statement, so could someone please help me get through this. Thank You. P.S. This can be proved through elementary analysis results instead of going into ...
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0answers
20 views

Open valued and lsc correspondence

If $F$ is a correspondence which is open-valued and lower semicontiuous, then Graph of $F$ is open. This is what I tried so far; Given any point $(x,y)$ in Graph $F$, I am trying to find a ...
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2answers
50 views

Proving that $f(x) \le\ g(x)$, for all $x \in\ [c,b)$ [closed]

Let $f$ and $g$ be functions which are differentiable on $(a,b)$ with $$f(c) = g(c)$$ for some $c\in (a,b)$. If $$ f'(x) \le\ g'(x),\qquad \forall \;x \in [c,b) $$ prove that $$ f(x) \le ...
4
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1answer
68 views

Fake proof: Equivalence of norms

Good morning. I'm having a hard time finding what's wrong with the following argument. Let $f$ be any function in $C^{1}([0;1])$ and let $||f||$ and $N(f)$ be two norms defined as follows: $$||f|| = ...
2
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2answers
38 views

Topology - function is continuous definition clarifications

In Rudin - Real and complex Analysis we have written: If $X$ and $Y$ are topological spaces and if $f$ is a mapping of $X$ into $Y$, then $f$ is said to be continuous provided that $f^{-1}(V)$ is an ...
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1answer
55 views

Is the function $f: (1, \infty) \to \mathbb R$ defined as $f(x)=\sum_{n=1}^\infty n^{-x}$ continuous ?

Is the function $f: (1, \infty) \to \mathbb R$ defined as $f(x)=\sum_{n=1}^\infty n^{-x}$ continuous ? I know that for each $n \ge 1$ , the function $g:(1,\infty) \to \mathbb R ; g(x)=n^{-x}$ is ...
2
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1answer
82 views

If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$?

Let, $id:(X,d_1)\to (X,d_2)$ is continuous. Then which is(/are) TRUE ? (A) $X$ must be singleton. (B) $X$ can be any finite set. (C) $X$ can NOT be infinite (D) $X$ may be infinite but NOT ...
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2answers
53 views

Is this Epsilon-Delta approach to prove that $e^x$ is continuous correct?

I couldn't find an epsilon-delta proof for continuity of $e^x$ so here's my take: Suppose $|x - x_0| < \delta$ and fix $\epsilon >0$ Consider $|e^x - e^{x_0}| < \epsilon$, then ...
2
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1answer
33 views

Showing a mapping is a Homeomorphism

I am trying to prove that the Stone Cech Compactification map is a homeomorphism. I have most the proof finished, but I am stuck on showing that the inverse function is continuous. Here is what I have ...
2
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0answers
49 views

Uniform convergence and equicontinuity

Given a sequence of functions which is not uniformly convergent, can we deduce, that none of its subsequences is uniformly continous and therefore, by Arzela-Ascoli say that the family of function is ...
0
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1answer
23 views

I need help proving this problem using the Intermediate Value Theorem?

I only need help with part (a). I figure once I get that, part (b) should be easy. Anyway, I know I'm supposed to let $h$ be a function $h(t) = f(t) - g(t)$, and use the IVT from this point to ...
0
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1answer
33 views

Prove that an absolutely continuous cdf is continuous

Let $F(x_1,\ldots,x_d)$ be an absolutely continuous distribution function. How to prove that $F$ is continuous? Thank you.
1
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1answer
34 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 ...
4
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1answer
47 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} ...
2
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3answers
60 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
70 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 $v(\gamma=0+, \rho)=v(\gamma=0-,\rho)$ ...
0
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1answer
35 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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1answer
336 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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1answer
62 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
34 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 ...
2
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3answers
40 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 $$\epsilon>0$$ ...
0
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0answers
53 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
45 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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2answers
38 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
29 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
20 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 ...
0
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2answers
96 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 ...
1
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1answer
34 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 ...
0
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1answer
17 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 ...
0
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1answer
26 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 ...