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)
17,358
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Is function $f(x) = \frac{x-1}{x-1}$ defined at $x=1$? [duplicate]
I think the answer should be YES. Because the function is actually
$f(x) = 1$
Likewise, the function $f(x) = \frac{x^2-1}{x-1}$ is identical to $f(x) = x+1$. Therefore it is well defined at $x=1$.
Can ...
0
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0
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8
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Bump functions of maximal height under smoothness constraints
Suppose I want to find a function that is of maximal height at $0$ and of zero height outside of the unit ball. In other words, a bump function of maximal height.
The constraint I put on this function ...
- 2,455
1
vote
2
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49
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On the continuity of the function $f(x)=\int_1^\infty \frac{\cos t}{x^2+t^2}dt$.
Let $F(x)=\int_1^\infty \frac{\cos t}{x^2+t^2}dt$. Then which of the following are correct?
$f$ is bounded on $\mathbb R$
$f$ is continuous on $\mathbb R$
$f$ is not defined everywhere on $\mathbb ...
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votes
2
answers
58
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If the logarithm of two sequences gets close together, do the sequences get close together?
I should define what I mean by two sequences $(a_n)$ and $(b_n)$ that are close together. What I mean is that for any $\epsilon > 0$, $\exists N$ such that for $n \geq N$ we have $|a_n-b_n| < \...
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3
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1
answer
108
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$\int _0 ^x \frac {g(u)}{u+f(x)} du=1 $ then $f ' (0) =?$
Given
$\int _0 ^x \frac {g(u)}{u+f(x)} du=1 $ where $f$ and $g$ are continuous on $[0 , \infty ) , f >0 $ on $(0 , \infty)$ and $g>0 $ on $[0 , \infty )$ then $f'(0) $ is equal to
$(a)\space \...
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3
votes
1
answer
66
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Nondecreasing interval on a locally absolutely continuous function $f: [a, b] \to \mathbb{R}$ with $f(b) > f(a)$
For my research, I need to verify whether a function $f: [a, b] \to \mathbb{R}$ with $f(b) > f(a)$
has a nondecreasing interval $(c, d) \subset [a, b]$ such that $f$ is increasing on that interval ...
- 119
0
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0
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19
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Is this functional jointly continuous on the product space?
Let $\Omega : = [0,1]^2$ be our sample space and $\Delta \Omega$ the set of probability measures on $\Omega$. Let $\omega \in \Omega$ denote the typical element of $\Omega$.
Let $x := (x_1,x_2) \in X :...
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1
answer
46
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Prove/disprove: $d(x, \phi(x)) \leq f(x) - f(\phi(x))$ implies $\phi$ is a contraction
According to Approach0, this question seems new.
There is a similar question, however in that question asked for a proof about a function, which satisfied the definition (see Problem), has a fix point....
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33
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Suppose that $f(x)$ is continuous in $[0, 1]$ and $f(0)=0=f(1)$. Prove $f(c)=1−2c^2$ for some $c\in(0,1)$ [closed]
Please someone tell me how to solve this question and most importantly the reasons behind steps
i tried to solve it by making another function g(x) = f(x) - (1-2x^2)
but i don't know after this step ...
1
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0
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18
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Is the composition of a $ C^\infty $ function with a saturation function still $ C^\infty $?
I am studying the composition of functions and their differentiability properties. Specifically, I have a $ C^\infty$ function $ f $ and a saturation function defined as:
$$
\text{sat}(u) =
\begin{...
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0
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23
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Dini's theorem but this time $\{g\}$ sequence increases. [closed]
Every proof I've seen until now says that $\{f\}$ sequence is monoton. This means it can be increasing or decreasing. But they prove the theorem by assuming $\{f\}$ to be decreasing. And it seems like ...
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0
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1
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50
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Is it possible to construct a compact superset based on a closed continuous interval? [closed]
Let $I = [a,b] \subset \mathbb{R}$ be a closed interval and $D \subset \mathbb{R}^{n}$ be compact. Furthermore, $g : D \to \mathbb{R}^{n \times n}$ is a continuous function and $f: I \times D \to \...
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3
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63
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Map from $[-1,1]\to S^1$ that are continuous at exactly $1$ or $0$ points?
The following is a problem from a GRE review booklet:
Let $f$ be a function with domain $[-1,1]$, such that the coordinates of each point $(x,y)$ of its graph satisfy $x^2 + y^2 = 1$. The total number ...
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0
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3
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47
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Example of closure of image not equals image of closure under continuous mapping [duplicate]
Let $f:X\to Y$ be a continuous mapping, and let $E\subseteq X$, I have proved that
$$
f(\overline{E})\subseteq \overline{f(E)}
$$
But I got stuck when trying to find an example such that $f(\overline{...
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23
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Does there exist a continuous function that changes sign and monotonicity at the same point? [closed]
I'm looking for a function with the following properties, and I'm wondering whether such a function exists. Let $I$ be an open interval, and let $c \in I $ be a point within that interval. I would ...
0
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27
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Truncation of $C^k$ functions in $C^k$
When we truncate a function $f$ we usually consider $g_n(x)=f(x)\mathbb{1}_{[-n,n]}(x)+f(n)\mathbb{1}_{(n,\infty)}(x)+f(-n)\mathbb{1}_{(-\infty,-n)}(x)$, where $n$ is a natural number. Can we do it ...
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42
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Is $f$ continuous at $x = \frac 1 2$ if? [closed]
Let $f:\mathbb R \to \mathbb R$ be the function defined by: $f(x) = 1$, if $x \in \mathbb Q$; $f(x) = 0$, if $x \in \mathbb{R} \setminus \mathbb Q$.
Is $f(x)$ continuous at $x = \frac 12?$
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18
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Countable discontinuities preserved under uniform convergence.
Let's suppose we have a sequence of functions defined on a set $A$ a subset of $\mathbb{R}$, $f_n \rightarrow f$ where the convergence is uniform. Additionally, each $f_n$ has at most a countable ...
1
vote
1
answer
67
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Neighborhood of a point can get smaller and smaller?
Lemma
Let $I$ be a time interval, and for each $t ∈ I$ suppose we have two statements, a “hypothesis” $H(t)$ and a “conclusion” $C(t)$. Suppose we can verify the following four assertions:
a. If $H(t)$...
- 4,925
4
votes
1
answer
54
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Boundedness of the modulus of continuity
According to Wikipedia, the modulus of continuity is "used to measure quantitatively the uniform continuity of functions", and is defined as follows:
For a function $f: I \rightarrow \mathbf{...
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votes
1
answer
61
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Extensions preserving Lipschitz constant
Let $f:K \to \mathbb{R}$ be a $C^2$ function defined on the set $K=[0,1]^2$ (more generally, $K$ could be a convex and compact set in $\mathbb{R}^d$) such that $f$ and its derivatives are pointwise-...
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2
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70
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On real functions with the property that for each $\varepsilon>0,$ every interval contains two points whose secant line is $>\varepsilon.$
Let $f:\mathbb{R}\to\mathbb{R}$ have the following property:
$$\forall\varepsilon>0,\text{ no matter how large },\forall a<b,\ \exists a<x_1<x_2<b\text{ such that }\left\lvert\frac{f(...
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8
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1
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239
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Is a continuous real function with vanishing derivative in all but countably many points constant?
The Cantor function is a standard example of a function $f:[0,1]\to\mathbb{R}$ that is continuous, has almost everywhere zero derivative, and nonetheless is not constant. More specifically, the number ...
- 7,524
3
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1
answer
33
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Continuity and Surjectivity of Distance-Expanding Functions
Let $K\subseteq \mathbb R$ be non-empty and $f: K\to K$ be continuous such that $$|x-y|\leq |f(x)-f(y)|~\forall x,y\in K.$$ Which of the following statements are true?
$1.$ $f$ need not surjective.
$2....
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1
answer
33
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almost single-valued upper hemicontinuous function [closed]
Let $f$ be a set-valued function defined on $\mathbb{R}$.
It satisfies the following conditions:
(1) For any $t \in \mathbb{R}$, $f(t)$ is a closed and bounded subset of $\mathbb{R}$;
(2) $f(q)$ is a ...
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3
votes
2
answers
78
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show that $x\longmapsto \arg \min\|x-y\| $ is continuous
Let $\varnothing\ne M\subset\mathbb R^n$ be a convex compact set, suppose $M\subseteq B_r(u)$ (closed ball).
Then the map
\begin{align}
f:\ B_r(u)&\longrightarrow M
\newline x&\longmapsto f(...
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0
answers
43
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How to check the order of continuity of a function?
I was reading on the topic of order of continuity and came up with this question.
Is there a maximum $k$ such that a function $f:\mathbb{R}\to\mathbb{R}$ is in $C^{k}$ but not in $C^{k+1}$?.
I was ...
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2
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1
answer
76
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$ f:X\to Y $ is continuous on $ X $ iff $ \forall A\subset X $, $ f(\bar{A})\subset \overline{f(A)} $. [duplicate]
I have always thought this result as a generalised version of the sequential criterion of continuity. Surely, $ \bar{A} $ contains all the convergent sequences from $ A $, and the result says that the ...
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3
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0
answers
63
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Doubts over the Second fundamental theorem of calculus
I've read in many places two different definitions of the second FTC, I'm interested about the general form:
As presented here
If $F$ is differentiable on $[a,b]$ and the derivative $F'=f$ (say) is ...
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2
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0
answers
27
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Summability of integrand and continuity of integral function
Problem
Let $f:[0,\infty) \to (0,+\infty)$ a continuous function such that
$$\sup_{t>0}\{t^n f(t)\}<+\infty, \quad \forall n\in \mathbb{N}.$$
$(a)$ Prove that the map $$t\mapsto t^{x-1}f(t)$$ is ...
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1
answer
64
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Do piecewise continuous functions form an inner product space?
While studying Fourier series, I encountered the fact that
The set of piecewise continuous functions along with the inner product $$\langle f,g \rangle=\int_0 ^{2\pi}f(x)g(x) \, \mathrm{d}x$$ form an ...
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1
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55
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On the continuity of a function, again
Sorry to bother you with another question on continuity, but I need experts to explain me things. I cannot interact with books only.
So the question is this: we know a definition of continuity thought ...
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47
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Showing the family $\{x^n/n \}_{n\in\mathbb{N}}$ is equicontinuous on $[0,1]$
As the title implies, let $\{x^n/n \}_n\subseteq C[0,1]$, where $x\in X$ and $X$ is a metric space. I want to show that this family is equicontinuous at each $x\in[0,1]$. By the definition then, we ...
6
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2
answers
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How many solutions does $z^{\sqrt{2}}=2$ have?
Let's explore:
$$f(z)=z^{\sqrt2}-2$$
Question
How many solutions does $f$ have in the complex plane?
Finite, Countable or Uncountable?
Exposition
One may define $$e^{a+bi}=e^{a}(\cos(a)+i\sin(b)), \...
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1
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Is this measure of how insincere my path marker is always differentiable?
The formal question:
Given $f(x)$, a continuous-but-not-necessarily-differentiable-everywhere function.
Define $g(x)$ as:
$$g(x) = \begin{cases}
0 & \text{if } x \leq 0 \\
\int_0^x \max(f(x) - f(...
- 922
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votes
0
answers
63
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Relation between continuous function and regular space
Let $T,Y$ topological spaces with $Y$ a regular space, $X \subset T$ a dense subspace and $f: T \to Y$ a function with the property that $f\vert_{X \cup \{t \}}$ is continuous for all $t \in T$. Prove ...
3
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1
answer
76
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How to prove that the improper integral is uniformly continuous?
Let $f$ be a uniformly continuous and bounded function on $\mathbb{R}$, and let $g$ be a continuous on $\mathbb{R}$ such that $\int_{-\infty}^\infty |g(x)| dx$ converges. Show that $$F(x)=\int_{-\...
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2
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102
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Prove by epsilon delta method that $f(x) = \log(x)$ is continuous at $x=2$ [duplicate]
I am confused, about how to proceed with this question, I got this far only and got stuck :
For any $\varepsilon>0$, there exist $δ>0$ such that $$|\log(x) - \log(2)|<\varepsilon $$whenever
$$...
- 9
2
votes
2
answers
127
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Continuity without topology nor $\epsilon$-$\delta$
I was reading this question, and the related answers, which popped out in merit to my previous question about continuity while searching over here: $\lim_{x\to 0} f(x)$ where $0$ is isolated in the ...
- 265
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1
answer
52
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Why is "going faster around a circle" not a homotopy from the identity map on $S^1$ to a constant map?
Let $S^1 \subseteq \mathbb{C}$ be the unit circle, and consider the map $f : S^1 \times [0, 1] \to S^1$ defined for $\theta \in [0, 1)$ and $t \in [0, 1]$ as:
$$f(e^{2\pi i\theta}, t) = \cases{
e^{\...
- 734
1
vote
1
answer
56
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Continuity in the topological sense, and singlets as open/closed sets
First of all I apologise if this question will sound stupid. I'm approaching to the study of topological spaces and real analysis in a deeper way, that is trying to fill holes and doubts and this one ...
- 265
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votes
2
answers
113
views
Problem in mathematical analysis
Prove that the function $f(x) = x|\sin(x)|$ gets in the interval
$(0,\infty)$ every positive value infinite times.
In other words prove that for every $0<y$ the $f(x) = y$ equation have infinite ...
0
votes
0
answers
21
views
Non-linear, continuous, strictly increasing, unbounded function with a constant average rate of change?
I have a strictly increasing, continuous, non-linear, unbounded (above and below) function $w:\mathbb{R}\rightarrow{} \mathbb{R}$ such that for fixed $x,y\in \mathbb{R}$, it holds that for some ...
- 1
1
vote
1
answer
41
views
Continuous functions on compact Hausdorff space with a special property.
Let $X$ be a compact Hausdorff space and $- : X \to X$ be a homeomorphism of $X$ with period 2. i.e, $\bar{(\bar{t})} = t, \forall t \in X$.
Define $C(X,-) := \{ f \in C(X,\mathbb{C}): f(\bar{t})=\...
- 151
0
votes
1
answer
56
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continuous function on $\mathbb{R}^*$
I am having a confusion, what I know is that the function $f(x) = \dfrac{1}{x}$ is continuous on $\mathbb{R^*}$, since $\mathbb{R^*}$ is its domain.
But when I was discussing this with one of my ...
-1
votes
1
answer
44
views
Show that if a subset of $C([0,1])$ is open relative to one norm it is also open relative to another or vice versa
I am struggling coming up with a solution to the following question. Consider the two norms $||f||_1 = \int_0^1 |f(x)| \mathrm{d}x$ and $||f||_\infty = \sup_{x \in [0, 1]} |f(x)|$ on $C([0,1])$ (the ...
3
votes
0
answers
60
views
Proving differentiability at a point - Confusion? [duplicate]
(For a more fleshed-out example demonstrating my confusion, see this post. In John's answer, why do we have the "smoothness" condition? Wouldn't this be necessary if and only when dealing ...
- 1,021
1
vote
1
answer
63
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Why is this sufficient for continuity?
I don't understand why the following argument suffices for continuity in the proof of the following statement:
Let $X,Y,Z$ be topological spaces and consider $\mathscr{C}(Y,X) := \{ f: Y \rightarrow X:...
- 927
0
votes
0
answers
38
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Proof of a continuity theorem for parameter-dependent integral functions
I try to prove the following theorem about the continuity of a parameter dependent integral function.
Thm. Let $(X,\mathcal{M})$ a measurable space, $\mu:\mathcal{M} \to [0,+\infty]$ a positive ...
- 951
1
vote
0
answers
39
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Continuity of spectrum as multiset
Let $n$ be an integer, and let $D$ be the set of diagonalizable matrices with complex coefficients. Let $S$ be the space of cardinal $n$ multisubsets of $\mathbb{C}$, that is, it is the quotient space ...
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