Skip to main content

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)

Filter by
Sorted by
Tagged with
3 votes
1 answer
38 views

Defining matrix function through its diagonalization

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a smooth function. Let $Dg(n)$ be the set of diagonal $n \times n$ matrices, and define $\overline{f}:Dg(n) \to Dg(n)$ in the following way: first, for a ...
ImHackingXD's user avatar
  • 1,076
0 votes
0 answers
7 views

Continuity regarding integration over indicator function

i have problems proving the following: Let $D\subset\mathbb{R}^d$ be an open, bounded and simply connected set. Let $f:D\rightarrow\mathbb{R}$ be a continuous and bounded function with $|\nabla f(x)|&...
lleon97's user avatar
0 votes
1 answer
63 views

Continuity of the evaluation map defined on $X \times C(X,Y)$

I'm currently reading chapter 7 of the book Topology by Munkres and was wondering if my "proof" for one of the theorems works. In particular, it is the theorem that states that if $X$ is a ...
A P's user avatar
  • 1
0 votes
0 answers
32 views

(Non)existence of a continuous map $\mathbb{RP}^{2k-1}\to\mathbb{RP^{2k-1}}$ with no fixed points

I would like to check my solution to a question that asks whether there is a continuous map $f:\mathbb{RP}^{2k-1}\to\mathbb{RP}^{2k-1}$ with no fixed points. I think that there is no such map. Brouwer'...
utx7563yu's user avatar
1 vote
1 answer
27 views

Show that the given map is continuous in a Hausdorff space

Let $E$ is an $n$-dimensional Hausdorff topological vector space and $F$ is any topological vector space. If, $\{e_1, \cdots,e_n \}$ any basis for $E$ and $T: E \to F$ is linear map, then show that $$...
emily_everdeen's user avatar
-1 votes
2 answers
71 views

Derivative bounded below and preimage of compact set is compact

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable, and assume that $\vert f'(x) \vert \ge 1$, for all $x \in \mathbb{R}$. Then for each compact set $C \subset \mathbb{R}$, the set $f^{-1}...
dhashtp's user avatar
  • 51
0 votes
0 answers
21 views

Why does Lipschitz continuity imply Holder continuity?

I was wondering why would a Lipschitz continous function also be Holder continous? Shouldn't the implication be backwards, because Lipschitz continuity is a special case of Holder continuity where the ...
Riccardo Caiulo's user avatar
1 vote
1 answer
30 views

Explanation for proof of the continuity of the function in $t$ defined as the integral of a continuous function of two variables

Statement: Let $f:[a,b]\times\mathbb{R} \to \mathbb{R}, (x,t) \mapsto f(x,t)\in\mathbb{R}$ be a continuous function. Then $$ F(t) := \int_a^b f(x,t)dx $$ is continuous. I understand the proof for this ...
Lucas's user avatar
  • 19
-2 votes
0 answers
57 views

Continuity of the function $f(x)=x+\frac{1}{x}$ at $1$ [closed]

Prove that the function $x+\frac{1}{x}$ is continuous at $p=1$. Just to add some context: before this question from the book I'm reading, it's given that $|f(x)-f(1)|\leq(1+\frac{1}{x})|x-1|$ for $x&...
Batata's user avatar
  • 55
0 votes
0 answers
29 views

What is meant by f being continous in possible isolated points in its domain?

The definition of a continuous function in my textbook is given as follows: $f$ being continuous in a accumulation point $a\in D_f$ means that: $\lim_{x\to a}f(x)=f(a)$ We count $f$ as continuous in ...
user3612's user avatar
0 votes
1 answer
46 views

Fixed point of continuous function $[1,10 ]$ to $(2,8)$

Statement: Every continuous function $f:[1,10] \to (2,8)$ has a fixed point. True/ False I feel it is True, since image of $[1,10]$ will be a compact subset in $(2,8)$, by fixed point theorem we must ...
dhashtp's user avatar
  • 51
-1 votes
0 answers
52 views

Differentiability implies continuity because we wouldn't get an indeterminate form? [closed]

Related : Why isn't continuity part of the definition of differentiability? Instead of conjugation or whatever like here or here, why can't I argue like this? For $f:A \to \mathbb R$, let $a \in ...
BCLC's user avatar
  • 13.6k
2 votes
0 answers
26 views

Minimums of 2-D function form a continuous function itself. [closed]

Let $B_{\delta}\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B_{\delta}\to \Bbb{R}_{\geq 0}$ be real-analytic and have only one zero in $B_{\delta}$, ...
Doofenshmert's user avatar
0 votes
0 answers
35 views

Suppose that $\{f_{n}\}$ converges uniformly to $f$, and let $g_{n}(x)=f_{n}(x+1/n)$. Show that $\{g_{n}\}$ converges to $f$.

This problem is from my calculus textbook about uniform and pointwise convergence. Problem. Suppose that $\{f_{n}\}$ converges uniformly to $f$, and let $g_{n}(x)=f_{n}(x+1/n)$. Show that $\{g_{n}\}$ ...
legogubben's user avatar
0 votes
1 answer
42 views

how to verify that $f$ is uniformly continuous on $(0,+\infty)$ [closed]

define $f=x^\alpha$, where $0<\alpha<1$, my question is, how to verify that $f$ is uniformly continuous on $(0,+\infty)$ I have finish the proof when $\alpha=$$\frac{1}{2}$, since the fraction ...
EddyLiu's user avatar
1 vote
0 answers
74 views

Possible values of the integral of the function with the information of derivative

Let $f: [0,1] \to \mathbb{R}$ denote some continuously differentiable function such that $f '(x)$ is continuous and $ \int_0^1 f(x)dx=0$. If $\max_{x\in[0,1]}|f'(x)|=24$, then what are some/all of the ...
Looping outlaw's user avatar
0 votes
3 answers
120 views

prove that the function $f(x) = x + \frac{1}{{x^2}}$ is continuous on the interval $(2, \infty)$ using epsilon delta definitions

Let $f(x) = x + \frac{1}{{x^2}}$ and I want to prove that $f$ is continuous at a point $c$ in the interval $(2, \infty).$ I started by defining $\delta$ as $\min(1, \text{ })$ and then I tried to ...
Nicholas Gray's user avatar
0 votes
1 answer
27 views

Reconciling Continuity of Binary Relations with Continuity of Functions/Correspondences

I asked this question in the Economics StackExchange as well, but figured it may be better-suited here. There are various ways to express the concept of continuity of a binary relation, but one I've ...
hillard28's user avatar
0 votes
0 answers
113 views

$f(x) \ge 0$ for all $x \in [a, b]$ and $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$.

Suppose that $f$ is continuous on $[a, b]$, that $f(x) \ge 0$ for all $x \in [a, b]$ and that $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$. My attempt: Let $\dot{\Pi}$ be a tagged ...
user13's user avatar
  • 1,679
0 votes
2 answers
43 views

Showing a certain type of function on $\mathbb R ^d$ is Lipschitz?

I want to prove the following: Assume that $f:\mathbb R ^d \to \mathbb R$ is continuous, convex and $|f(x)|\leq a+b|x|$. Then $f$ is Lipschitz. I thought it would follow immediately from the ...
uniform_on_compacts's user avatar
0 votes
1 answer
35 views

Prove symmetry of probabilities given random variables are iid and have (not absolutely) continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous. Denote common distribution as $$F(y) :...
BCLC's user avatar
  • 13.6k
1 vote
1 answer
39 views

Prove independence of events given random variables are iid and have (not absolutely) continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_Y(y) := F_{Y_1}(y) = F_{Y_2}(y)...
BCLC's user avatar
  • 13.6k
1 vote
1 answer
60 views

Prove the following result on continuity.

Continuity of Eigenvalues Let $A\in \mathbb C^{n,n}$ and $\lambda \in \sigma(A)$ with algebraic multiplicity $m$. Then ($\forall \epsilon>0$)($\exists \delta >0$)($\forall E\in\mathbb C^{n,n}, ||...
Unknown x's user avatar
  • 783
2 votes
1 answer
66 views

Pointwise limit of continuous functions whose graph is in a given closed set

Let $C\subseteq\mathbb R^2$ be a closed set with the property that for every $x\in\mathbb R$, there exists at least one $y\in\mathbb R$ such that $(x,y)\in C$. Does there exist a function $f:\mathbb R\...
triple_sec's user avatar
  • 23.5k
0 votes
1 answer
22 views

Show that the expression defines a norm

Let us consider the real vector space $V := C([-1, 1])$. Show that the expression $|| f || := max_{x ∈ [-1,1]}abs(\frac{1}{2}f(x))$ (for any $f ∈ V$) defines a norm on $V$. To show that the expression ...
Катерина Ковальова's user avatar
-2 votes
1 answer
67 views

How can I prove that if $f:[0,+\infty)\to[0,1]$ is continuous and increasing, then it is uniformly continuous? [closed]

The problem is the following: Let $f:[0,+\infty)\rightarrow\mathbb{R}$ such that it is continuous, strictly growing and its image is contained in $[0,1]$, is uniformly continuous. I've tried in ...
Fausto Martinez's user avatar
0 votes
0 answers
20 views

Existence of all partial derivatives implies anything about total derivative’s existence?

Say a function $f: \mathbb{R}^n \to \mathbb{R}^m$ is $C^{\infty}$ if its partial derivatives of all orders exist. Does this imply anything about the existence of its total derivatives, or if not then ...
Abced Decba's user avatar
0 votes
0 answers
15 views

Semidifferentiability at the extremum of an interval and continuous extension of derivative

Let $f:[a,b] \to \mathbb{R}$ be differentiable on $(a,b)$ with continuous derivative $f'$. (i) Assuming that $f'$ can be continuously extended at $a$, is it true that $f$ is semidifferentiable at $a$ ...
Paolo Intuito's user avatar
2 votes
1 answer
47 views

Local extreme points of $f(x)=\frac{5}{4} x^{\frac{4}{5}} - |x-2|$

Consider the real function $f(x)=\frac{5}{4} x^{\frac{4}{5}} - |x-2|$. $f$ is a continuous function and it is differentiable on $\mathbb{R}$ except at $\{0,2\}$. Indeed, $f$ is as follows: (i) If $x \...
user237522's user avatar
  • 6,539
1 vote
2 answers
91 views

Necessary and sufficient condition for $f(f(x))=x$

Based on the concept that $f$ is increasing then all the solutions of the equation $f(x)=f^{-1}(x)$ lie on the line $y=x$. I have two questions: Can I solve $f(f(x))=x$ by taking $f(x)=x$ if $f$ is ...
Vikas Sharma's user avatar
4 votes
1 answer
69 views

Proving a (Representing Utility) Function is Continuous

I sincerely apologize for posting such a long question. The question involves a complicated proof of a theorem in mathematical economics. I feel it will be better for me to state my question first. I ...
Beerus's user avatar
  • 1,919
2 votes
1 answer
51 views

Continuous basis $(e_1(t),e_2(t))$

Assume that $t \mapsto A(t)$ is a $n \times n$ matrix-valued continuous function and that we know that $\ker A(t)$ has dimension $2$ for every real $t$. Prove that there is are continuous vector-...
J.Mayol's user avatar
  • 790
1 vote
2 answers
63 views

Ambiguity in solving differential equations

Suppose we want to solve the differential equation $y'=x \sqrt{y}$. Easy right? Because you can transform the equation into a separable one. However, I think that there are more than meets the eye. ...
legogubben's user avatar
0 votes
1 answer
46 views

Existence of composite function with algebraic and exponential functions

We are supposed to test the validity of two statements: (1) There exists a differentiable function $g:R\rightarrow R$ such that $g(x^3+x^5)=e^x-100$ (2) There exists a continuous function $g:R\...
Equiposied's user avatar
2 votes
2 answers
56 views

Show that there does not exist any holomorphic function on the open unit disk and continuous on the closed unit disk with the given property.

Let $\mathbb D : = \left \{z \in \mathbb C\ :\ \left \lvert z \right \rvert < 1 \right \}.$ Prove that there is no continuous function $f : \overline {\mathbb D} \longrightarrow \mathbb C$ such ...
Anacardium's user avatar
  • 2,472
0 votes
1 answer
37 views

Mistake in solution to show that $\frac{x^ay}{x^2+y^2}$ is continuous at $(0,0)$ if $a>1$.

I was working through some practice problems, and I'm not sure where I'm wrong in the following proof. The function $g:\mathbb{R}^2 \rightarrow \mathbb{R}$ is defined by \begin{cases} \frac{x^ay}...
Rudinable's user avatar
1 vote
0 answers
81 views

A question regarding the deformation retraction of a ball without origin to a sphere.

Let $D=\{||x||\leq 1\}\subset \mathbb R^d$ be a ball or radius $1$ with center at origin $O$ and let $H:\overline (D\setminus\{O\})\times[0,1]\to D\setminus\{O\}$ be a strong deformation retraction to ...
Mircea's user avatar
  • 564
1 vote
1 answer
52 views

How to show the given function is differentiable...?

Consider the function $f:\mathbb{R^2}\to \mathbb{R}$ defined by $$\begin{equation}f(x,y)=\begin{cases}(x-y)^2\sin \frac{1}{x-y},&\text{ if } x\ne y\\0 ,&\text{ if } x=y\end{cases}\end{equation}...
math student's user avatar
  • 1,245
0 votes
1 answer
31 views

Find a constant so that the limit at 1 exist

I thought this problem was going to be "easy" but I do not know what I am missing. The problem gives you this function $f(x) = \begin{cases} \frac{\sin(\pi x)}{\ln{x}} &\...
MiguelCG's user avatar
  • 339
2 votes
4 answers
89 views

How to prove the sequence of function $\{f_n\}$ is Cauchy but not convergent? (Details Below)

Problem Consider the space $C[-1,1]$, together with the norm defined by $\|f\|_1 = \int_{-1}^1|f|d\lambda$ (where $\lambda$ is the Lebesgue measure). For each $n$ define a function $f_n:[-1,1]\to\...
Beerus's user avatar
  • 1,919
0 votes
1 answer
23 views

Why $(g\circ f)(\omega)$ must be compact if $g$ is continuous....

Let $\omega =\cup_{i=1}^5(i,i+1)\subset \mathbb{R}$ and $f:\omega \to \mathbb{R}$ be differentiable such that $f'(x)=0\forall x\in \omega$. Let $g:\mathbb{R}\to \mathbb{R}$ be any function, then which ...
math student's user avatar
  • 1,245
9 votes
3 answers
448 views

Circular definition of continuity

When evaluating limits, it's tempting to just plug in the approach value into the function to get the answer. For example, if $f(x) = x^2$ and we need to solve $\lim_{x \to 2}(f(x))$ then we want to ...
Lauren S's user avatar
  • 339
4 votes
3 answers
585 views

A strange confusion over a problem of continuity in Multivariate Calculus.

For $\beta\in\Bbb R,$ define $$f(x,y)= \begin{cases}\cfrac{x^2|x|^{\beta}y}{x^4+y^2},&x\neq 0 \\ 0, &x=0 \end{cases}$$ Prove that at $(0,0)$ the function is discontinuous if $\beta=0.$ My ...
Thomas Finley's user avatar
1 vote
1 answer
75 views

Given that $f(x) = f(\frac{x}{1-x}) \forall x \ne 1$, and $f$ is continuous at $0$. Find all such $f$.

Question Given that $f(x) = f(\frac{x}{1-x}) \forall x \ne 1$, and $f$ is continuous at $0$. Find all such $f$. Attempt Substituting $x$ with $\frac{1}{n}$, I got that $f(\frac{1}{n}) = f(\frac{1}{n-1}...
Debu's user avatar
  • 656
0 votes
0 answers
34 views

Proving $g(x) = 1 − \frac{\lVert x−p\rVert}{\delta}$ is continuous

I want to show that given any two points $p, q \in K$ with $p\neq q$ we can choose a continuous function $g \in C(K)$ so that $g(p)\neq g(q)$, specifically by letting $g(x) = 1 − \frac{\lVert x−p\...
jet's user avatar
  • 467
0 votes
0 answers
24 views

Dominated Convergence Theorem and Norm convergence

Here is the theory I have in mind: Dominated Convergence Theorem: $(X,A,\mu)$ measure space, $g$ a $[0,\infty]$-valued integrable function on $X$, $f,f_1,f_2,\dots$ $[-\infty,\infty]$-valued $A$-...
Waaal's user avatar
  • 309
5 votes
3 answers
161 views

If $f(x): [0,4]\to\{-1,1,0\}$, find the number of functions $f(x)$ that are discontinuous at every integer and continuous at all other points. [closed]

I tried: If the intervals $(0,1)$ and $(1,2)$ are taking the same values, we have $3\cdot 2$ ways to set the integer $1$ and its 'surroundings', and then if they have different values then $C^3_2\cdot ...
Frenzy Ripper's user avatar
0 votes
3 answers
71 views

Medium Hard continuity proof

Let $c: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function whose derivative $c^{\prime}$ is continuous at 0 with $c^{\prime}(0)=1$. (a) Use the continuity of $c^{\prime}$ at 0 to prove $\...
TanWu's user avatar
  • 13
0 votes
0 answers
36 views

Translation in $L^{\infty}$

Consider the translation operator $\tau_h$ defined on $L^\infty(\mathbb{R}^n)$ s.t. $\tau_hu(x)=u(x-h)$. I know that $\tau_h$ is not continuous with respect to $h$, I mean it’s not true that $h\to 0$ ...
Shiva's user avatar
  • 81
0 votes
1 answer
62 views

Suppose that exists a $M>0$ s.t. $|f(x)-f(p)| \leq M|g(x)-g(p)|$ for all $x$. Prove that if $g$ is continuous at $p$, then $f$ is as well.

"Let $f$ and $g$ be defined in $\mathbb{R}$ and suppose that exists a $M>0 \hspace{0.2cm}(\exists)$ st. $|f(x)-f(p)| \leq M|g(x)-g(p)|$ for all $x$. Prove that if $g$ is continuous at $p$, ...
Batata's user avatar
  • 55

1
2 3 4 5
344