Questions tagged [continuity]

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) or (general-topology)

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. A map $f : X \to Y$ is said to be continuous at $x_0$ if for every $\varepsilon > 0$, there is $\delta > 0$ such that $d_Y(f(x_0), f(x)) < \delta$ whenever $d_X(x_0, x) < \varepsilon$. A map $f : X \to Y$ is said to be continuous if it is continuous at $x_0$ for all $x_0 \in X$.

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A map $f : X \to Y$ is said to be continuous if $U \in \tau_Y$ implies that $f^{-1}(U) \in \tau_X$.

In the case of metric spaces, the metric induces a topology, and the two notions of continuity coincide. Note that multiple metrics can induce the same topology, and that not all topologies are metrizable (can be generated from some metric).

Continuity is a sufficient condition for the intermediate value theore. It is necessary for the extreme value theorem, as well as differentiability.

15514 questions
4
votes
3 answers

Uniform continuity of $f(x) = x\cos(x)$

Is it uniformly continuous? $ f: [0,\infty] $ $$f= x\cos(x)$$ I proved that if $$|x_1 -x_2| \rightarrow 0 \quad \mathrm{then}\quad |f(x_{1})-f(x_{2})| \rightarrow 1 $$ so it is not uniformly continuous, right?
aiki93
  • 867
  • 6
  • 11
4
votes
1 answer

For fixed $n \in \mathbb{R}$, what are all the continuous functions that satisfy $nf(x) = f(nx)$?

For fixed $n \in \mathbb{R}$, what are all the continuous functions that satisfy $nf(x) = f(nx)$? I thought it would just be functions of the form $f(x) = kx$ but, for example, in the $n=2$ case we have that $f(x) = x \cos (\frac{2\pi \ln x }{…
mtheorylord
  • 4,432
  • 12
  • 38
4
votes
1 answer

Intermediate Value Theorem and Fundamental Theorem of Calculus question

I received a question on a previous exam, but I had no clue how to go about doing it. I know I'm supposed to use the MVT, IVT and FTC, but I'm not sure where. The question is Suppose $f(x)$ is integrable on $[a,b]$, with $f(x)\geq0$ on $[a,b]$,…
4
votes
2 answers

Discontinuous linear operator

Let $X = C ^ \infty ([0,1] , \mathbb R )$ and let $\|\cdot\|$ be any norm of $X$. Define the operator $T:X\to X$ by $T(f) = \frac{df}{dx}$. Show that $T$ is not a continuous linear operator from $( X , \|\cdot\| )$ into $(X , \|\cdot\| )$,…
4
votes
2 answers

Proof Verification: Prove $\sqrt{x}$ is uniformly continuous on [0, $\infty$)

Proof: Fix $\epsilon \gt 0$. We want to find $\delta \gt 0$ such that: $$|x-a| \lt \delta \Rightarrow |\sqrt{x} - \sqrt{a}| \lt \epsilon$$ $$\phantom{2000i11111}\Rightarrow |\frac{x-a}{\sqrt{x}+\sqrt{a}}| \lt \epsilon$$ If $|x-a|\lt \delta$ then…
user438530
4
votes
0 answers

Tough question on Lebesgue measure and the distance function

Here is a problem I cannot get my head around. Let $E⊆\mathbb{R}$ be Lebesgue measurable with $\lambda(E)>0$. Consider the distance function $d(x)=\inf\{|t-x|:t\in E\}$. Prove $\lim_{x\rightarrow y}\frac{d(x)}{|x-y|}=0,$ a.e. $y\in E$. My…
MelaniesWoes
  • 709
  • 4
  • 15
4
votes
4 answers

Show that f is continuous on x if and only if x is irrational

Let $f : [0, 1] → R$ be given by the formula $f(x) = \frac{1}b$ if $x = \frac{a}b$, where $a$ and $b$ have no common factor, $f(x) = 0$ if $x$ irrational. Show that $f$ is continuous at $x$ if and only if $x$ is irrational. (Hint: Use the density…
zodross
  • 303
  • 2
  • 8
4
votes
2 answers

Local connectedness is preserved under retractions

I want to show that if $X$ is a locally connected topological space, $A\subseteq X$ is a subspace and $f:X \rightarrow A$ is continuous such that $f|_{A} = Id_{A}$, then $A$ must be locally connected as well. My progress so far: Take $U\subseteq A$…
4
votes
2 answers

Let $f:\Bbb{R}\to \Bbb{R} $ a continuous function s.t. $f(x+y)+f (x-y)=2f(x)+2f(y) $. Find $f$.

Let $f:\Bbb{R}\to \Bbb{R} $ a continuous function s.t. $f(x+y)+f (x-y)=2f(x)+2f(y) $. Find $f$. My idea: Let $x=y$. Then $f (2x)+f(0)=4f(x)$. I tried to use this recurrence but it doesn't work.
rafa
  • 1,252
  • 5
  • 12
4
votes
1 answer

Usage of Schwarz Reflection Principle to Study Conformal Equivalence of Annuli

Let $A(1,r) = \{z \in \mathbb{C} : 1 < |z| < r\}$. I would like to prove the standard result that $A(1,r)$ and $A(1,r')$ are conformally equivalent iff $r = r'$. To prove the nontrivial direction, suppose that we have an analytic isomorphism $f…
Ashvin Swaminathan
  • 2,757
  • 10
  • 26
4
votes
1 answer

Give an example of a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies all three conditions

Give an example of a function $f:\mathbb{R} \to \mathbb{R}$ that satisfies all three conditions $f$ is bijective $f'(0)=0$ the inverse function $f^{-1}$ is not continuous at $0$ Does $f(x)=x^3$ satisfy the all three conditions? If not can any…
user546987
4
votes
1 answer

Function with continuous components, Zariski topology

I am trying to understand how the Zariski topology is different from the usual topology on the affine space $A^n$. Let $X$ be an affine algebraic subset of $A^n$, the $n$-dimensional affine space over $k$. Let $f_i:X\rightarrow A^1$, $i=1,...,k$ be…
Jiu
  • 1,377
  • 7
  • 18
4
votes
1 answer

Proving that continuous function $f:[0,2] \to \mathbb{R}$ such as $f(0)=f(2)$, $\exists x \in [0,1]:f(x)=f(x+1)$

I am having trouble formalizing the answer to this question. Any help would be appreciated. Is it true that for every continuous function $f:[0,2] \to \mathbb{R}$ such as $f(0)=f(2)$, $\exists x \in [0,1]:f(x)=f(x+1)$? If the function is periodic…
4
votes
4 answers

Discontinuity of $\sin(\frac{1}{x})$

I’ve heard and read in many books that the function $$\sin\left(\frac{1}{x}\right)$$ is discontinuous at $x=0$ since as $x$ tends to zero the function ‘oscillates’ rapidly that is , for numbers very close to each other the number takes valued such…
Aditi
  • 1,351
  • 10
  • 26
4
votes
2 answers

Prove function is not differentiable even though all directional derivatives exist and it is continuous.

In the following link, the function below is provided as an example of a function being continuous and all directional derivatives existing. Yet, it is not differentiable. How do I prove that this is not differentiable? $$ f(x,y)= \begin{cases} …
Rohit Pandey
  • 5,457
  • 2
  • 22
  • 44
1 2 3
99
100