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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)

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Uniformly continuous and continuous functions one-to-one and onto

For a function that goes between two metric spaces, are uniformly continuous and continuous functions one-to-one? Are they on-to?
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25 views

Property takagi function by induction.

In a lecture on Applied Functional Analysis, the professor showed us some properties of the Takagi function from this paper. He wrote at the end the following property and said it could be easily done ...
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1answer
35 views

Proving a single function continuous

I am supposed to check the following function for continuity: $$h(x) = \frac {x^5}{x+5}$$ Knowing that the limits approaching from both sides have to be equal for the function to be continuous, I ...
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2answers
35 views

Proving functions as continuous

I was tasked with proving these two functions are continuous: $$g(x)=\begin{cases}3+\log(x), & x<e\\5, & x\ge e\end{cases}$$ $$j(x)=\begin{cases}x\sin\left(\frac{1}{x}\right), & ...
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1answer
37 views

Is there a way to mathematically prove $\psi (\mathbf{r})$ varies continuously (using the intuitive arguments provided below)?

Electric potential at a point outside the charge distribution is: $\displaystyle \psi (\mathbf{r})= \int_{V'} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ where: $\mathbf{r}=(x,y,z)$ ...
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23 views

$A$ is continuous in $0_X$ => $\exists c \in \mathbb{R}$, $ c \geq 0$ : $||Ax||_Y \leq c ||x||_X $

the map $A : X \rightarrow Y $ is linear, where $(X,||.||)$ and $(Y,||.||)$ are normalized vector spaces. I already have a solution, which is correct but a friend of mine showed me his solution and ...
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3answers
89 views

Is composition of surjective continuous function with discontinuous function discontinuous?

Let $I_1,I_2,I_3$ be intervals $\subset \mathbb{R}$. Suppose $f:I_1 \to I_2$ is a surjective continuous function and $g: I_2 \to I_3$ is a discontinuous function. Must the composition $g \circ f$ be ...
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1answer
32 views

Every isometry is Lipchitz-continuous

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, $f:(X,d_X) \rightarrow (Y,d_Y)$ be an isometry. Then $f$ is Lipchitz-continuous. Attempt: Suppose that $f$ is an isometry. Then for all $x_1,x_2$ in $...
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1answer
58 views

Show that $f(x) = {1\over x^n}$ is continuous in its domain, $n\in\Bbb N$

Let $n\in\Bbb N$. Show that $$ f(x) = {1\over x^n} $$ is continuous in its domain. I've recently shown that $g(x) = x^n$ is continuous everywhere in $\Bbb R$. Now I want to do the same for $1/x^n$,...
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1answer
24 views

Total Variation of Subintervals

Studying functions of bounded variation, the following exercise showed up: Let $I = [a,b]$ be an interval, $(E, \| \cdot\|)_E$ a normed vector space and $f \in BV(I, E)$ a function which is ...
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1answer
47 views

A Lipschitz function is $C^1$?

I am wondering if a Lipschitz function $f:[a,b]\to\mathbb{R}$ is $C^1$, that is its derivative is also continuous? I have seen that in a text however I could not prove it and does not seem so obvious ...
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1answer
50 views

Convergence/Divergence speed of $u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ given $f, g$ continuous and non-negative

Let be $f, g : [0, 1] \to \mathbb{R}_{+}^{*}$ continuous maps such that: $\forall n \in \mathbb{N}, u_n = \int_0^1 f(x)^n g(x) \textrm{d}x$ I want to show that $v = \left(\dfrac{u_{n + 1}}{u_n}\...
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3answers
89 views

Prove that there exists $c$ such that $f'(\xi)=\frac{f(c)-f(a)}{c-a}$ or $f'(\xi)=\frac{f(c)-f(b)}{c-b}$

Let $f(x) \in C^2 [a,b]$ and $f''(x) \neq 0$. Prove that for any $\xi \in (a.b)$ , there exists $c \in [a,b]$ such that $$f'(\xi)=\frac{f(c)-f(a)}{c-a}\qquad \text{or} \qquad~f'(\xi)=\frac{f(c)-f(b)}{...
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Proving that there is a $\theta\in(13\pi/6 , 7\pi/2 )$ s.t. $\tan(2\theta - 5\pi)\tan(3\theta + 4\pi) = 2/3$

I found this question while working , Proving that there is a $\theta\in (13\pi/6, 7\pi/2 )$ such that $$\tan(2\theta - 5\pi)\tan(3\theta + 4\pi) = 2/3$$ I started by defining a function on $\...
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1answer
55 views

Theorem 30.1 (b) in Munkres' TOPOLOGY, 2nd ed: The sequential criterion for continuity

Here is Theorem 30.2 in the book Topology by James R. Munkres, 2nd edition: Let $X$ be a topological space. (a) Let $A$ be a subset of $X$. If there is a sequence of points of $A$ converging ...
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2answers
35 views

How to prove the continuity? [duplicate]

Prove that the function : $f(x) = x$ when $x$ is rational and $f(x) = 1 - x$ when $x$ is irrational is continuous at $x = \frac{1}{2}$ To prove the continuity it is necessary to prove that for every $...
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4answers
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Prove $\frac{b-a}{6}≤\sqrt{1+b} - \sqrt{1+a}≤\frac{b-a}{4}$ if $3\leq a<b\leq 8$ [on hold]

I don't really know if I should use brute force or some kind of theorem, it comes on a calculus past exam and it says: suppose: $3≤a<b≤8$ prove that $$\frac{b-a}{6}≤\sqrt{1+b} - \sqrt{1+a}≤\frac{b-...
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2answers
37 views

Prove that $f: X \rightarrow R$ is continuous with respect to the metric $d_1$ on $X$ iff it is continuous with respect to the metric $d_2$ on $X$.

Let $X$ be a set and $d_1, d_2$ be metrics on $X$ so that for constants $m,M > 0$ and any $x,y \in X$ we have $md_1(x,y) \leq d_2(x,y) \leq Md_1(x,y)$ Prove that $f: X \rightarrow \mathbb{R}$ is ...
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1answer
55 views

Differentiability of $|x|^p$?

Let $p > 0$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined piecewise by $f(x)= |x|^p$ if $x \in \mathbb{Q}$ and $f(x)=0$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. For what values of $p$...
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0answers
43 views

Let $f:[a,b] \rightarrow \mathbb{R}$ is cts with $f(a), f(b) < 0, \int_a^b f(x)dx \ge 0$. Show $\exists c \in (a,b) s.t. \int_c^b f(t) dt \ge 0$.

My solution to this problem is the following: Solution: $[a,b]$ is closed interval, so $f$ is uniformly continuous on the interval $[a,b]$. So, we know that there exists $\delta >0$ such that ...
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1answer
16 views

Left-hand limit and Right-hand limit of a function

There is a function given by $$f(x)=\begin{cases} x\sin{\frac{1}{x}}, & x \ne 0 \\ 0, & x=0. \end{cases}$$ Find the left-hand limit and right-hand limit and the continuity of this function at $...
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1answer
21 views

Check the existence of partial derivatives and continuity of $f(x,y)$ at $(1,0)$ : $f(x,y) = \frac{3y(x-1)} {{({x-1})^2} +y^2}$

Check the existence of partial derivatives and continuity of $f(x,y)$ at $(1,0)$ : $f(x,y) = \frac{3y(x-1)} {{({x-1})^2} +y^2}$ when $(x,y)\neq (1,0)$ and $0$ otherwise. I decided to check continuity ...
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2answers
42 views

Let $f:K \rightarrow N$ be a continuous function from a compact $K$. Show that $f$ is uniformly continuous

I'm having trouble finishing this. One approach that I made is this: Let $\epsilon > 0$. Then, since $f$ is continuous, for every $x \in K$ exists $\delta_x > 0$ such that $d(x, x')<\delta_x ...
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1answer
27 views

continuity functions proof

siven $f : [a,b]\to \mathbb{R}$ continuous such that $f(a)\ne f(b)$ , prove that $\exists c\in(a,b)$ such that $$f(c)=\frac{f(a)+f(b)}{2}$$ I have only thought of using Intermediate value theorem , I ...
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2answers
59 views

Is this function $f: \mathbb{R}^{n+1}\rightarrow{S^{n}}$ continuous?

Let $f: \mathbb{R}^{n+1}\setminus{\{0\}}\rightarrow{S^{n}}$ and $g: S^{n}\rightarrow{\mathbb{R}^{n+1}}\setminus{\{0\}}$ be functions given by $f(x)=\displaystyle\frac{x}{||x||}$ and $g(x)=x$, where $||...
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1answer
78 views

Proving that $x/(x+1)$ is continuous at $x=2$ by the definition

I came upon this question, the function $f(x)=\frac{x}{x+1}$ is continuous at $x=2$ by the definition. Do I have to show that limit at $x=2$ exits first? I am confused with the steps, can someone ...
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0answers
28 views

Show that a càdlàg function is uniformly right-continuous on compact intervals

Let $(E,d)$ be a locally compact separable metric space, $I\subseteq\mathbb R$ be an interval, $f:I\to E$ be càdlàg and $a,b\in I$ with $a<b$. How can we show that $\left.f\right|_{[a,\:b]}$ is ...
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1answer
32 views

Discuss the function for continuity at (0,0) [closed]

$$f(x,y)=\begin{cases} 0,& (x,y)=(2y,y)\\ \exp[|x-2y|/(x^2-4xy+4y^2)],& (x,y)≠(2y,y).\end{cases}$$ Please give full answer to help me understand .
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1answer
14 views

Classification of image (in interval) of polynomial (non constant)

There is something I am trying to prove: Let $f:\mathbb R\to\mathbb R$ be a nonconstant polynomial. Show that the image of the function is either the real line, $[a, \infty)$, or $(-\infty,a]$ for ...
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0answers
41 views

A question about boundedness, continuity, and integrability

Didn't include my drafting but not sure if my answer is right.
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2answers
41 views

Open and closed sets and continuous functions

Suppose that $(X,ρ)$ is a metric space, $f:(X,ρ)\rightarrow R$ a continuous function and $D$ a dense subset of $X$ so as $f(D)$ finite. Prove that: (i) The range $f(X)$ is finite (ii) For every $t\...
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1answer
52 views

Prove that limit does not exist [on hold]

Let $f$ be a bounded function and continous in $[a, \infty )$ $M=\sup(f(x)), m=\inf(f(x))$ when $x\in [a, \infty )$ and suppose that $M,m$ are not in image of $f$. Prove that limit $\lim_{x\to\infty}(...
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2answers
37 views

Prove that $f$ gets its maximum

Let $f$ be a continous positive function in $(0,1)$ and $\lim_{x\to 0^+}(f(x))=\lim_{x\to 1^-}(f(x))=0$ Prove that $f$ gets its maximum in (0,1)
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0answers
9 views

Name for cases where “preimage map” preserves some property

Let $f^{-1} : 2^Y \to 2^X $ denote the "preimage map" associated with $f$ where $f$ is a function from $X$ to $Y$ and $2^X$ denotes the powerset of $X$ . There seem to be a few cases where a ...
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1answer
146 views

Let $f(x)$ be continuous from $[0, +\infty)$ to $ [0, +\infty)$, and $\int_{0}^{+\infty}f(x)dx$ diverges. [duplicate]

Prove there exists some $a>0$ such that the series $\sum_nf(an)$ diverges. I think it can be useful to partition $[0, +\infty)$ on $[n, n+1)$ and choose some $a_n$ for every $n$. But I can't ...
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1answer
31 views

Rationals can be the set of continuity of a function? [duplicate]

Most of the functions that I have seen have their discontinuities on rationals and continuities on irrationals! I am wondering if there is any exampe of some function whose continuities are rationals?...
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1answer
58 views

What sets can we define continuity and differentiability on?

I am an undergraduate Physics student (completing my first year shortly) who has had a (first) course on Calculus, and another on Linear Algebra. When working with differential equations (in physical ...
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1answer
35 views

Continuous then measurable?

I have this proposition: Prop. Every continuous functions $f:\mathbb{R}^n \to \mathbb{R}$ is $\mathcal{B^n} - \mathcal{B}$ measurable. I assume here $\mathbb{R}, \mathbb{R}^n$ count with their ...
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1answer
50 views

Proof continuous function $g$ defined on $[0,1]$ has a fixedpoint $x \in [0,1]$ [duplicate]

Claim: For the continuous function $g:[0,1]\rightarrow[0,1] \exists x \in [0,1]: g(x_1) = x_1$ Proof: Case 1: If g(0) = 0 and/or g(1) = 1 then g has at least one fix point on one of the ends of the ...
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2answers
50 views

Let $f: [0,10) \to [0,10] $ be a continuous function then which is correct

Let $f: [0,10) \to [0,10] $ be a continous map then (a) $f$ need not have any fixed point (b) $f$ has atleast $10$ fixed point (c) $f$ has atleast $9$ fixed point (d) $f$ has atleast one fixed ...
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1answer
32 views

Does there exist a homotopy between identity function and any continuous function?

(My question is related to the Brouwer fixed-point theorem.) Let $B$ be a closed ball of $\mathbb{R}^n$. Q 1. If $f : B \rightarrow B$ is a continuous function, then is there a homotopy between $...
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0answers
24 views

Suggestion for seminar about rings of continuous functions [closed]

I have to do a seminar about the rings of continuous functions, it will be a part of a course in topology. The main topic of my seminar will be the functor from the topological space and the ...
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1answer
49 views

Continuity of function defined by integral

Define $F:[0,\infty)\to\mathbb{R}$ by $$F(t)=\int_{0}^{\infty}e^{-tx}\dfrac{\sin x}{x}\,dx.$$ Prove that $F$ is continuous at $t=0$. I already proved that $$F'(t)=-\dfrac{1}{1+t^2}, \quad t>...
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2answers
38 views

If $f$ is lipschitz, then $|f(x)| < C \left(1+|x|^λ\right)$

I'm in a first course of analysis and we got this question and I wasn't able to figure it out. Any hint's are welcome. If $f$ is lipschitz, then $\vert f(x)\vert<C(1+\vert x\vert ^λ)$ for some $C,\...
3
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1answer
56 views

Is this function continuous at x=2?

I have a function, k, below that I think is not continuous at x=2, but I'm not sure. If it is how can it be proven. Let: $h(x) = \frac{x^2+20}{6} $ $g(x) =\frac{12+8x-x^2}{6}$ $t(x) =4+\frac{2}{3}(...
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2answers
30 views

monotonic function can only have simple discontinuity

I am self-studying Rudin, Principles of Mathematical Analysis. I am having trouble going through the theorem saying that monotonical functions can only have simple discontinuity, i.e., Suppose $f$ is ...
0
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1answer
37 views

Difference between Limits and Continuity

I know that it may be a repeated question but that question failed to give me a satisfactory answer.Can someone provide with me an easy expiation for the question below I know limit of function ...
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2answers
57 views

max f$( (a,b) )$= max $f( [a,b] )$ in continuous function

Let $f:[a,b]→\mathbb{R}$ be a continuous function and gets its maximum in $(a,b)$. Prove that max $f( (a,b) )$= max $f( [a,b] )$
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2answers
30 views

Continuous function in [0,1] for each x f(x)>x [closed]

Continuous function in [0,1] for each x f(x)>x, Prove that exists c>0 for each x in [0,1] f(x)>x+c.
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1answer
35 views

$\lim_{n \to \infty} n \int_{0}^{100}f(x)g(nx)dx=f(0)$

Let $g: \mathbb{R} \to \mathbb{R}$ be a continuous function with $g(y)=0$ for all $y \notin [0,1]$ and $\int_{0}^{1}g(y)dy=1$. Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function. ...