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 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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15 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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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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0answers
15 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
19 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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63 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
54 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
40 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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4answers
49 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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16 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
62 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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1answer
39 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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1answer
17 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
14 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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0answers
40 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
32 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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3answers
57 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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2answers
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
18 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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3answers
71 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
33 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
33 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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2answers
104 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
87 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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252 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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52 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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1answer
31 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 ...
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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?
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0answers
17 views

Is the given function absolutely continuous

Define , $f:[0,1]\to \mathbb R$ by $$f(x)=\begin{cases}x\cos\frac{\pi}{2x}&\text{ for } x\not =0\\0&\text{ if } x=0\end{cases}$$ Then, $f$ is absolutely continuous or NOT ? I know that the ...
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0answers
35 views

$E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?

Problem The graph $G$ of $f$ is defined as the points $(x, f(x))$ for $x \in E$. Suppose $E \subset \mathbb{R}$ is compact, then $f : E \to \mathbb{R}$ is continuous iff its graph is compact. ...
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1answer
34 views

Prove that ${f_n}$ converges uniformly to f [duplicate]

Let $f:\mathbb{R}→\mathbb{R}$ be uniformly continuous; let $\{y_n\}⊂\mathbb{R}$ be such that $\lim y_n=0$; and define the sequence $f_n:\mathbb{R}→\mathbb{R}$ by $f_n(x)=f(x+y_n)$. Prove that ${f_n}$ ...
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2answers
48 views

Sufficient rigor in proving $f(x)$ is continuous at the origin, for $f$ analog to the Dirichlet function.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function given by: $$ f(x)=\begin{cases} 2x,&x\in\mathbb{Q}\\ -2x,&x\in\mathbb{I}\end{cases} $$ $f(x)$ is continuous at the origin $0$. Proof: ...
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1answer
47 views

Problem 8 (a) in Exercises after Sec. 18 in Munkres' Topology, 2nd ed.: How to show this set is closed? [duplicate]

Let $X$ be an arbitrary topological space, let $Y$ be an ordered set in the order topology, and let the maps $f, g \colon X \to Y$ be continuous. Then how to show that the set $S$ given by $$S \colon= ...
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1answer
62 views

Prove $f(x) =0$ for all $x \in [a, b]$ [duplicate]

If $f:[a,b] \rightarrow R$ be continuous , let $f(x) =0$ when $x$ is rational. Prove $f(x) =0$ for all $x$ that is an element of $[a,b]$. Thanks for the help I can't seem to find the f[a,b] notation ...
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1answer
46 views

Is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ?

Let $(X,d)$ be a metric space , then is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ? Do we need completeness of $X$ ?
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If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal?

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal ? The thing is , since $X$ is finite , so it is compact , so ideal $M$ is maximal iff it is of the form ...
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2answers
50 views

If $u_n \rightarrow u$ in $C([a, b])$, is it true that $\int u_n(x) dx \rightarrow \int u(x) dx$?

Let $u_n \rightarrow u$ in $C([a, b])$.Is it true that $$ \underset{a}{\overset{(a+b)/2}{\int}} u_n(x) dx \rightarrow \underset{a}{\overset{(a+b)/2}{\int}} u(x) dx ?$$ If $u_n \rightarrow u$ in ...
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1answer
24 views

How to get tietze extension theorm (for metric spaces) with arbitrary co-domain of real valued function

I know the tietze extension theorem on with bounded range namely " If $F$ is a closed subset of a metric space $X$ such that $f:F \to [a,b]$ is a real valued continuous function , then there is a ...
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1answer
29 views

Show that a linear form $\mathbb{R}^n \to\mathbb{R}$ is continuous

$f(x)$=$n∑k=1$ $g$($x_k$) ou $x_k$ is the kth component of the vector x. $x_k=\langle e_k,x\rangle$. I have the option of showing this with sequences (which I dont know how, I never understood how to ...
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2answers
29 views

Proving the continuity of a function with two variables using partial derivatives

Let $$f(x,y) = \left\{\begin{array}{rcl}\frac{x^{\alpha}y^{\beta}}{x^2+y^2} & \mathrm{if} & (x,y) \ne (0,0)\\ 0 & \mathrm{if} & (x,y)=(0,0)\end{array}\right.$$ The question is: for ...
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1answer
38 views

Esoteric question on discontinuities, ln(x)?

Suppose I have ln(x), the domain is given as x > 0, range is all reals. Now suppose I asked for the points of discontinuity of ln(x). How would one answer this question? Is there an infinite ...
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2answers
31 views

Continuity of functions of several variables

trying to understand this example of continuity of a a function on $R^2$ $$f(x,y) := \begin{cases} \frac{xy^2}{x^2 + y^2}& \text{if } (x,y) \neq (0,0)\\ 0 & \text{if } ...
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3answers
53 views

Continuity of a piecewise function at a specific point

I am having trouble proving the following function is not continuous at $x = 0$ using a formal definition of continuity. $ f(x) = \left\{ \begin{array}{lr} \sin(\frac{1}{x}) & : x \neq 0\\ ...
3
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1answer
69 views

Can a function be continuous at one value of $x$ and discontinuous at all other $x\in\mathbb R$

I have: $$f(x)=\begin{cases} x,&\text{if }x\in\Bbb Q\\ 0,&\text{if }x\notin\Bbb Q\;, \end{cases}$$ This function is continuous at $x=0$, but discontinuous everywhere else? (Between $f(0)=0$ ...
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0answers
23 views

Trying to prove Tietze extension theorem

I am trying to prove Tietze extension theorem for metric spaces that is " If $X$ is a metric space , $F$ is a closed set in $X$ and $f:F \to [0,1]$ is a continuous function , then there is a ...
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2answers
36 views

Connectedness of $\{(x,\sin(\frac{1}{x})); x \in ]0,1]\}$

Let $A = \{(x,\sin(\frac{1}{x})); x \in ]0,1]\}$ I need to show that $A$ is connected. I am trying to use the following theorem: If $(X,d_1)$ and $(Y,d_2)$ are two metric spaces, and $f: X ...
0
votes
1answer
28 views

Cauchy sequence under a uniform continuous function

Let , $f:(1,4)\to \mathbb R$ be uniformly continuous and $\{a_n\}$ be a Cauchy sequence in $(1,2)$. Consider: $x_n=a_n^2f(a_n^2)$ and $y_n=\frac{1}{1+a_n^2}f(a_n^2)$ Then which is ...