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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10 views

Continuity over Basis for the neighborhood

I'm trying to solve the following exercise: Let $f: X \rightarrow Y$ be a function from a metric space $X$ into a metric space $Y$. Let $a \in X$ and let $B_{f(a)}$ be a basis for the neighborhood ...
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Bounded Variation and Continuity

Let $f$ be of bounded variation on $\mathbb{R}$. Let $$ V_{f}(x)=V(f ;-\infty, x) $$ Prove that $f$ is continuous at $c$ if and only if $V_{f}$ is continuous at $c$. Update: $V(f ;-\infty, c)=\sup \{V(...
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Let the complex hilbert space be $H=\mathcal{L}^{2}[0,\frac{\pi}{2}]$. Consider application T and prove the following.

\begin{equation} \forall t\in \left[0,\frac{\pi}{2} \right] \qquad \qquad Tf(t)=\cos (t)\int_{0}^{t}\sin(s)f(s)ds \end{equation} Show that $ Tf $ is a continuous function with $ f \ in H $. ...
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Proof verification for “continuity of function $f\implies $ continuity of $m( x) =\min_{a\leq t\leq x} f( t)$”.

Let $\displaystyle f:[ a,b]\rightarrow \mathbb{R}$ be a continuous function, that is $\displaystyle f\in \mathcal{C}[ a,b]$. It is to be proven that $\displaystyle m\in \mathcal{C}[ a,b]$, where $\...
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Let f be a given continuous constant function on a compact metric then how I approximate f by a function which attains sup at exactly one point?

I tried in many way but can't solve I tried to approximate it by the reciprocal of sum of constant one and distance function but that is not an approximation. Can anyone plz help.
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18 views

Integrability and continuity

I know that an continous function is integrable and also a monotone and bounded function is integrable. But if a function is only bounded and integrable is not necessary that it is continous right?
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3answers
168 views

Show that: $f(x)$ is a constant function

Let $f(x)$ be a continuous function on $\mathbb{R}$, such that for any real number $x$ we have: $$\lim_{h\to 0}\dfrac{1}{h^3}\int_{-h}^{h}f(x+t)\cdot t\,dt=0.$$ Show that: $f(x)$ is a constant ...
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1answer
35 views

Is this an equivalent definition of Hölder continuity?

I am wondering if this is an equivalent definition of Hölder continuity for a real function $f:\mathbb{R}\rightarrow \mathbb{R}$: Assume that there exists $\Delta,\alpha,K>0$ such that if $|x-y|<...
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2answers
34 views

How to show that a continuous function analytic in a deleted neighbourhood is analytic in the entire neighbourhood?

Suppose $$ F(z)= \begin{cases} \dfrac{f(z)-f(a)}{z-a}, & z\ne a\\[2ex] \quad f'(a), & z=a \end{cases} $$ Here $f$ is analytic on a simply connected domain $D$ containing $a$. Clearly, $F$ is ...
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1answer
26 views

Non-continuous addition in metric space

Is there an example of a metric d on some linear space X, which makes the addition of vectors a non-continuous operation? Where we endow the product space with the metric $d((x, y), (x', y')) = d(x, x'...
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Is there a continuous function defined on $\mathbb{R}$ which is a bijection on $\mathbb{R}\backslash\mathbb{Q}$ but not a bijection on $\mathbb{Q}$.

Is there a continuous function defined on $\mathbb{R}$ which is a bijection on $\mathbb{R}\backslash\mathbb{Q}$ but not a bijection on $\mathbb{Q}$. I tried to argue in this way. Let $p,q\in\mathbb{Q}...
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1answer
71 views

Given a continuous $f : \mathbb{Q} \to \mathbb{Q}$ is there a cont $g:\mathbb{R} \to \mathbb{R}$ such that restriction of $g$ to $\mathbb{Q}$ is f?

Given a continuous $f : \mathbb{Q} \to \mathbb{Q}$ there is a cont $g:\mathbb{R} \to \mathbb{R}$ such that restriction of $g$ to $\mathbb{Q}$ is f The example I could come up with is $$f:\mathbb{Q} \...
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Linearized problem related to the proof of open

Recently, I read the paper of K.-S. Chou & X.-J. Wang. In the proof of Theorem A, I was confused. It says Let’s consider (1) $$\det(h_{ij}+h\delta_{ij})=fh^{p-1} \tag{1}$$ where $f > 0$ in $C^...
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1answer
231 views

Does there exist such a continuous function $f:\mathbb R\to\mathbb R$, such that $f(f(f(x))) = e^{-x}$? [closed]

The problem's statement: Does there exist such a continuous function $f$ that $$ f(f(f(x))) = e^{-x} $$ Couldn't come up with anything that would use the function's continuity.
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Show that for $n \geq 2, f(x) = x^{1/n}$ is continuous on $[0, \infty).$

Here is some scratch work i did. Let $a \in (0, \infty).$ The case for which $n=2$ is easy to check. Now assume $f(x) = x^{1/k}$ is continuous on $[0, \infty)$ and let $\varepsilon > 0$. Then $\...
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2answers
47 views

Graph of $\cos^{-1}\left(\sqrt{1-x^2}\right)$

Plot the Graph of $f(x)=\cos^{-1}\left(\sqrt{1-x^2}\right)$ Sol: We see that $$f'(x)=\frac{d}{dx}\cos^{-1}\left(\sqrt{1-x^2}\right)=\frac{-1}{\sqrt{1-(1-x^2)}}\frac{d}{dx}\sqrt{1-x^2}$$ $\implies$ $$f'...
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26 views

Find a sequence of continuous functions that converges to a piecewise function

Let $f(x)= \left\{ \begin{array}{lc} 0, & x \leq 0 \\ \\ 1, & 0 < x \\ \end{array} \right.$ Find a sequence of $f_{n}$ such each one of $f_{n}: \...
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2answers
31 views

Do $C_c(I)$ and $C(I)$ coincide for an open interval $I$ and are the functions in $C_c(I)$ even uniformly continuous?

Most probably this question boils down to my confusion about the correct definition of $C_c(\Omega)$, but: If $\Omega$ is a topological space and $\operatorname{supp}f:=\overline{\{x\in\Omega:f(x)\ne0\...
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1answer
47 views

Is the function $f: \mathbb R \rightarrow \mathbb R$ defined by $f(x) = \exp(1/\sqrt{x^2-1})$ continuous in a topological sense?

I've recently started learning about topology, and I was able to prove that any function which is undefined on a countable subset of $\mathbb R$ is also not continuous. However, I am struggling to ...
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Show that a set is equicontinuous and uniformly bounded

Show that the set of functions S= {$\frac{sin(nx)}{n}$}$_{n≥1}$ on $I=[0,1]$ is equicontinuous and uniformly bounded. I'm not really sure how to go about this, could anyone offer any help?
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Infimum + continuous Question

Suppose the function $g : \mathbb{R}\to\mathbb{R}$ is continuous and strictly decreasing on $\mathbb{R}$ with $g(0) = 4, g(1) = 1$ and $g(2) = 0$. I've already solved the following problem: Explain ...
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1answer
16 views

By intermediate value theorem i can say there are n-1 roots which are strictly increasing. What can be said about other two roots?

Let $f(x)$ be a polynomial of degree $n+1$, with positive leading coefficient. Suppose $x_1<x_2< ..... <x_n $ such that { \begin{array}{ll} f(x_i) < 0 & \mbox{if } i \ is \ even \\ ...
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26 views

Is there a basis for sequences?

There are various equivalent definitions of a topological space and for some of them we have the concept of basis: a basis of opens sets, or a basis of neighbourhood. This concept simplifies verifying ...
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1answer
51 views

Show that a function of two variables is continuous at origin

Define $$f(x,y) = \begin{cases} \frac{x^2 {|x|}^\beta y}{x^4 + y^2}, & \text{if $x \neq$ 0} \\[2ex] 0, & \text{if $x = 0$}, \end{cases}$$ where $\beta>0$. By using epsilon delta definition,...
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1answer
33 views

Are all local martingales continuous in probability?

We say $X$ is continuous in probability if for any $t_0 \geqslant 0$ fixed, we have $ \mathbb{P}\left(\left\|X_{t}-X_{t_0}\right\|>\varepsilon\right) \rightarrow 0$ as $t \rightarrow ...
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1answer
39 views

If two continuous finitely-integrable functions of two variables differ at one point, does at least one of the single integral functions also differ?

Say we have two real-valued functions $p$ and $q$ of two variables, $x$ and $y$. They are continuous, with finite integral $\int p(x,y) dx dy < \infty$, $\int q(x,y) dx dy < \infty$, and they ...
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2answers
52 views

Finding out whether 3 given functions exist

Introduction I am trying to find links between continuity, the existence of one-sided limits in specific points and the existence of a function in a specific point. So I came up with these scenarios ...
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1answer
38 views

Thinking about continuous functions as vectors with continuum-many components.

Recently, I've been thinking about the vector space $C[a,b]$. I understand there's a lot out there about bases for this vector space, but what I want to try to understand are two things: Is the ...
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$f(k) = \int_0^{\pi/2} \frac{dx}{\sqrt{1-k\cos^2x}}$ then as a function of $k$ ; $f(k)$ is

If $$f(k) = \int_0^{\pi/2} \frac{dx}{\sqrt{1-k\cos^2x}}$$ $k$ belong to $(0,1)$ then as a function of $k$ ; $f(k)$ is A) Monotonic increasing B) Monotonic decreasing C) Non-monotonic D) Can not be ...
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1answer
12 views

Showing continuity of specific functions (relations involved)

Let $f : X \to Y$ be continuous. Let $R$ be equivalence relation on $X$ such that $xRy$ only if $f(x) = f(y)$ and $S$ be equivalence relation on $Y$. Let $g: X/R \to Y$ where $g([x]_R) := f(x)$ and $...
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how to design a function to capture the boundary condition for $u(x=-1,t)=u(x=1,t)$ (for a PDE)?

I need to come up with a boundary condition function for a PDE solving method desribed here: https://github.com/sciann/sciann-applications/blob/master/SciANN-BurgersEquation/SciANN-BurgersEquation....
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31 views

Extend continously an integral function

If I have an integral function, for instance $F(x)=\int_1^x \frac{1}{\sqrt{3-t}}\,dt$, surely the domain of the integrand is $(-\infty, 3)$ and then can I say that since $\lim_{x\to 3^-}F(x)=\int_1^{3}...
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Continous function transform equals zero, then function equals zero in $\mathbb{R}$

Im proving the following statement: Let $\alpha>0$, the gaussian weight $\omega(z)= \frac{\alpha}{\pi}e^{-\alpha \left | z \right |^2}$ is the unique continuous, radial weight verifying $\int_\...
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1answer
38 views

Show $\mathbf{f}$ is continuous at $(0,0)$ on every line through the origin but not is not continuous at $(0,0)$ for specific function - verificationn

Let $$f(x,y) = \begin{cases} 0, & |y| > x^{2}\ \text{or}\ y = 0 \\ 1, & \text{otherwise} \end{cases} $$ Show $\mathbf{f}$ is continuous at $(0,0)$ on every line through the origin but not ...
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1answer
57 views

Continuity and Differentiability at end point of an interval

Let $ f(x) = x(\sqrt {x} + \sqrt{x+1}) $ The problem had asked to check continuity and Differentiability at $ x = 0 $ First of all I noticed that the Natural Domain of the function is non negative ...
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1answer
31 views

A continuous function with a local maximum [duplicate]

Does that a continuous function $f$ of $R$ into $R$ has a local maximum at $(p,f(p))$ implies that there exists $\delta$ such that $f(x)$ monotonically increasing in $(p-\delta,p)$ and monotonically ...
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How can we discuss about the continuity of $\sin(1/x)$ using the sequential definition?

We know that $f (x) = \sin(1/x)$ is not continuous at $x = 0$. Hence, to prove that by sequencial definition, I thought of two sequences such that $x_n =1/x$ and $Y (x)=\sin x$. Now $x_n $ is ...
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34 views

Discontinuities in Anti-derivatives?

I've been having a look at whether antiderivatives have discontinuities when the integrand is a function that is piecewise continuous, or it has a finite number of discontinuities while still being ...
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Continuity in the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is as follows: $\frac{d}{dx}\int_{a}^{x} f(t)dt = f(x)$ I know that to integrate a function $f(t)$, it must be piece-wise continuous. However, I'm not too sure ...
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1answer
38 views

Difference between continuity point of a measure and of a function?

I was wondering what the relationship is between a continuity point of a measure and of a function? Or are they even related? A measure is of course a function in itself. Say we are taking the $\...
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18 views

Uniformly convergent sequence of continuous functions $f_n \to f$ on $(a,b)$, then $f$ is continuous?

Uniformly convergent sequence of continuous functions $f_n \to f$ on $(a,b)$, then $f$ is continuous. My proof like: Part 1: If the interval is closed $[a,b]$, by uniformly convergent, then $\forall \...
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Doubt in Step 3 of Theorem 7.32 in PMA Rudin regarding my way of apporach.

At first I never thought of bringing a new variable $y$ and instead defined functions $h_t(x)$ such that $h_t(x)=f(x)$ and $h_t(t)=f(t)$ but I realised this would not be general enough to prove this ...
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2answers
52 views

Monotonicity of quotient function from the monotonicity of original functions

Let, $h,g:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ be two continuous and increasing functions such that, $\frac{h(t)}{g(t)} \rightarrow 0$ as $t\rightarrow 0$. From this can we show that near 0, $\frac{...
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2answers
63 views

$g_{n}(x):=max \lbrace f_{1}(x),…\rbrace$ uniformly convergent if $\lbrace f_{n} \rbrace_{n \in \mathbb{N}}$ uniformly bounded and equicontinuous

Let $\lbrace f_{n} \rbrace_{n \in \mathbb{N}}$ a sequence of functions with real values which is uniformly bounded and equicontinuous in a metric space $X$. For each $n \in \mathbb{N}$, we define the ...
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1answer
32 views

Characterization of continuity

So I found that this: $$\lim\limits_{h \to 0}\|L(x+h)-L(x)\|=0$$ implied continuity. Intuitively, I'd say it means that wherever you approach x from the limit is $L(x)$ but I struggle to see why. How ...
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1answer
30 views

Picard-Lindelöf to show whether $u'=u^2-u^3$ has unique solution on an interval

This problem is taken from here. Consider the initial value problem: $$ u'=f(u,t)=u^2-u^3$$ $$ u(0)= 2/a>0 $$ $a$ is a small constant. How can I determine wheter a unique solution tom $t=0$ to $t=...
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1answer
82 views

Prove or disprove: $\lim_{x→0} f(x) = 0$

Let $f : \mathbb{R} → \mathbb{R}$ be a function such that for any $r ∈ \mathbb{R}$ , we have $$\lim_{n\rightarrow \infty }f\left ( \frac{r}{n} \right )= 0$$ Prove or disprove: $\lim_{x→0} f(x) = 0$ ...
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19 views

Universal approximation theorem for bag functions

Approximation Let $\mathcal{M}, \mathcal{T}$ be subsets of a topological space. $\mathcal{M}$ approximates $\mathcal{T}$ iff $\mathcal{T} \subseteq \overline{\mathcal{M}}$. Let $\sigma$ be ReLU. Let $\...
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0answers
29 views

Is $f$ twice continuously partially differentiable?

I have shown that a function is continuously partially differentiable. If it holds that $$\frac{\partial^2{f}}{\partial{y}\partial{x}}=\frac{\partial^2{f}}{\partial{x}\partial{y}}$$ forall $(x,y)\in \...
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1answer
16 views

continuity of general function with blocked partial derivatives

Let $f:\Bbb R^2\to\Bbb R$, with blocked partial derivatives at the neighborhood of $(0,0)$. I'm trying to formulate that f is continuous at $(0,0)$, I think it's should be easy argue that by Lagrange'...

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