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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29 views

Is it possible to define integration over a discontinous domain?

I'm trying to think about this from the Riemannian integration perspective so let me know if Lebesgue integration or something else is better. An example where I seem to be running into problems is ...
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Is there a function $f(x)$ that satisfies $f(e^x) = e^x f(x)$?

Does there exist any function $f$ such that $$f(e^x) = e^x f(x)?$$ If so, I believe it could be used to create an analytics versions of the function $\exp_{n+1}(x) = \exp(\exp_{n}(x))$ (where $n$ is ...
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Differentiable function for $\mathbb{Q} \rightarrow \mathbb{Z}$ [closed]

Are there any continuously (at least once) differentiable function that maps rational numbers $\mathbb{Q}$ to integers $\mathbb{Z}$? How about monotonic? Can you give me an example?
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show that a set is not closed

Let $$M:=\left\{ x+i \sin\left(\frac{1}{x}\right)\; |\; x \in (0,1) \right\}$$ not closed does not automatically mean open, right? For example, could I use the approach that every limit point of M is ...
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1answer
59 views

Proving discontinuity and continuity of a real valued function on an open interval.

Let $E\subset (a,b)$ be a countable subset of the open interval $(a,b)$. Let $E=\{x_n:n\in \mathbb N\}$. Let $\sum c_n$ be a convergent sequence such that $c_n\gt 0$ for all $n\in \mathbb N$. Let $f:(...
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1answer
26 views

Epsilon delta proof question

Define what it means to say that a function f : R → R is continuous at c ∈ R. Let f : R → R and g : R → R be functions that are continuous at 0, with f(0) > g(0). Prove that there is a δ > 0 ...
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17 views

Continuity and derivative of specific function

Define a function $f_{n}$ on $\mathbb{R}$ as $$ f_{n}(x)=\left\{\begin{array}{ll} 0 & \text { if } x=0 \\ x^{n} \sin \frac{1}{x} & \text { elsewhere } \end{array}\right. $$ where $n$ is a ...
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If the antiderivative of f is differentiable, then is f integrable? [closed]

If the antiderivative of f is differentiable, then is f integrable?
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42 views

Definition of continuity implies some kind of subset

Given $Z_n\rightarrow ^p b$, and if $g$ is continuous at $b$, then $g(Z_n)\rightarrow^p g(b)$. That's the context of the problem, but my real issue is this. The definition of continuity is $\forall\...
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Does the monotone pointwise limit of continuous functions has left and right limit everywhere?

Let $\{f_n(x)\}_{n\geq 0}$ be a sequence of continuous functions such that $f_n(x) > f_{n + 1}(x)$ for all $n\geq 0$. Assume that $f_n(x)$ converges pointwise to a function $f(x)$. It is clear that ...
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Seeking an example of continuous function that has no integrable derivative

I am seeking for a function $[0,1]\to \mathbb{R}$ that is continuous and has derivative almost everywhere, but this derivative is not integrable niether on sense of improper integrals. For instance, $...
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How does one arrive at this conclusion? [closed]

I can't figure out how to arrive at the correct answer for the below question. To me there seems no intuitive reason that the given option should be the correct one. Question: Let $f: [0,10]→[10,20]$ ...
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Infimum question. [closed]

This is the question. Could someone please show me the working? **Suppose the function g : R → R is continuous and strictly decreasing on R with g(0) = 4, g(1) = 1 and g(2) = 0. Explain why the set S =...
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Is such a function possible [duplicate]

Is it possible for a non-piecewise, non-gif function to have a value at a point but not be continuous at that point? (Eg. at x=2, f(2) exists, but the function is not continuous at x=2). Please ...
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31 views

Breaking a real valued increasing function into absolutely continuous, continuous and jump function

Suppose $F$ is an increasing function on $[a, b]$. (a) Prove that we can write $$ F=F_{A}+F_{C}+F_{J} $$ where each of the functions $F_{A}, F_{C}$, and $F_{J}$ is increasing and: (i) $F_{A}$ is ...
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Right continuity of the potential

Let $X=(X_t)_{t \geq 0}$ be a Hunt process with transition semigroup $(P_t)_{t \geq 0}$. We define for $\alpha >0$ and a bounded Borel-function $f$ the potential $U^{\alpha}f(x)=\int^{\infty}_0 e^{-...
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68 views

Let $f: \mathbb R \to \mathbb R$ be a strictly decreasing continuous function.

Let $f: \mathbb R \to \mathbb R$ be a strictly decreasing continuous function. Prove that there exists a real number a such that $f(a) = a$.
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Find $I_n(a)=\displaystyle \int \limits_0^1 \dfrac{dx}{(x^2+a^2)^n}, \, a\ne0, \, 0\ne n \in \mathbb{N}.$

Find $$I_n(a)=\displaystyle \int \limits_0^1 \dfrac{dx}{(x^2+a^2)^n}, \, a\ne0, \, 0\ne n \in \mathbb{N}.$$ I have following ideas. We have $I_n(a)=I_n(-a)$. \begin{align} \forall \,0\ne n \in \mathbb{...
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1answer
32 views

Find the values of $p$ and $q$ to make $f$ continuous

If $$f(x)=\begin{cases} x^2-1&-3\leq x<0\\ px+q&0\leq x\leq 1\\ x+1&1<x\leq 3 \end{cases}$$ is continuous on its domain. Then find the value of $p$ and $q.$ view question
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Continuity of a function from $\mathbb{R}^2$ to $\mathbb{R}$

Consider the function $f$ defined by $$f: \mathbb{R}^2 \to \mathbb{R} \ , \ (x_1,x_2) \mapsto \begin{cases} \frac{x_1^3x_2^2}{(x_1^2+x_2^2)^2} & \text{if} \ (x_1,x_2)\neq (0,0) \\ 0 ...
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Continuity in greatest integer function

$f(x)$= $(|x-2|)([x^2 -2x-2])$ where, [.]denotes the greatest integer function, then find the number of points of discontinuity in the interval $(\frac{1}{2},$2). Since, $|x-2|$ is continuous for all ...
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1answer
50 views

Let $f,g: \mathbb{R} \to \mathbb{R}$ be continuous functions and let $a$ be a real number such that $f(g(a)) = g(a)$ and $g(g(a)) = f(a)$.

Let $f,g: \mathbb{R} \to \mathbb{R}$ be continuous functions and let $a$ be a real number such that $f(g(a)) = g(a)$ and $g(g(a)) = f(a)$. Proof that there exists a real number $b$ such that $f(b) = g(...
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5answers
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The continuity of the function on $\mathbb{R}$: $I(y)=\displaystyle \int \limits_0^1sgn(x-y)\,dx $

Investigate the continuity of the function on $\mathbb{R}$: $$I(y)=\displaystyle \int \limits_0^1\text{sgn}(x-y)\,dx $$ I have some ideas: Considering $[a,b], \, \forall b>a>1$. Then, $\text{...
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Help showing that the open unit circle is homeomorphic to the half open plane.

The open unit circle $D=\{(x,y)| x^2+y^2<1\}$ and $H=\{(x,y)| y>0\}$ is homeomorphic to the open half-plane, prove it by building a function $(x,y):H\to D$ is a homeomorphism? It is often said ...
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3answers
133 views

Proving that $f: E\to \mathbb R$, defined by $f(x)=\frac{1}{x-x_0}$ is not uniformly continuous on $E$.

Let $E\subset \mathbb R$ and $x_0\notin E$ is a limit point of $E$. It is to be proven that $f: E\to \mathbb R$, defined by $f(x)=\frac{1}{x-x_0}$ is not uniformly continuous on $E$. I tried to prove ...
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1answer
37 views

How can I find whether the function f is differentiable and/or continuous?

The function that I have is: $$f(x)= \begin{cases}x^2 + 1 & \text{if } x \leq 0 \text{, and } \\ x^3 + 1 & \text{if } x > 0 \; .\end{cases}$$ For part a I have to find that f is ...
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1answer
18 views

Let $f$ be function from an open set $X\subseteq \mathbb{R}^{k}$ to $\mathbb{R}$ that are continuous on $X$

This is my first question so I hope I do it right. I got this question about a subset. Given that Let $f$ be function from an open set $X\subseteq \mathbb{R}^{k}$ to $\mathbb{R}$ that are continuous ...
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35 views

Modification of Lagrange’s Remainder Theorem to calculate $\ln(2)$

The following question is from Stephen Abbott's "Understanding Analysis." Question: Explain how Lagrange’s Remainder Theorem can be modified to prove $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} \...
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14 views

Stronger version of uniform continuity for continuously differentiable functions

We know that if $f$ is a continuous function on $[a,b]$ then it is uniformly continuous, that is, for any $\varepsilon>0$ there is $\delta>0$ such that $|f(x)-f(y)|<\varepsilon$ whenever $|x-...
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2answers
38 views

Possible values for continuous function?

Suppose $f$ is continuous on $[2,6]$, and the only solutions of the equation $f(x)=7$ are $x=2$ and $x=5$. If $f(3)=9$, then one of the following CANNOT be the value of $f(4)$ A) 9 B) 8 C)7.5 D)5 I ...
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1answer
27 views

Proving that this function is continuous based on definitions of two sets

Question: Let $f : \mathbb{R}^n \to \mathbb{R}$ such that $\{ \textbf{x} \in \mathbb{R}^n : f(x) > d\}$ and $\{ \textbf{x} \in \mathbb{R}^n : f(x) < d\}$ are both open sets for all real values ...
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0answers
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Looking for a sigmoid-like function with different properties

I am looking for a function that is 0 at 0, 1 at 1, increases to a predetermined $x_1$ relatively quickly, acquires a derivative close to 0 (but doesn't actually plateau) then starts increasing at $...
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18 views

If $f$ is continuous in $a$, then for all $u \in \mathbb{R}^n$ with $||u|| = 1, g_u$ is continuous in zero

Let there be a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and a point $a \in \mathbb{R}^n$. For all $u \in \mathbb{R}^n$ with $||u|| = 1$, define the following function: $g_u : \mathbb{R} \...
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Discontinuous set

We have a space $\mathcal{C}([0,1] \to \mathbb{R})$ of continuous functions on [0,1] and the uniform topology with the $sup$ metric. Define $\psi(f) = min\{1, \inf\{t \geq \frac{1}{2}: f(t) =0\}\}$. ...
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Prolonging the continuity of $g$

I have a doubt. If $$g(x,y) =\frac{(\sin(xy))}{(x+y)}$$ it's possible to prolong the continuity of this function to all points of the frontier of $g$? I know that I can prolong it to the origin if $(0,...
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2answers
26 views

Proof that a function with a discontinuity has Darboux Property

I have the following function: $$f:\Bbb R\to \Bbb R\text{ defined by } f(x)=\left\{\begin{matrix}0 & \text{ if } x\leq0, \\ \cos(x)\sin\left(\frac1{x}\right) &\text{ if } x>0. \end{matrix}\...
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1answer
68 views

How do I know if the ODE has a unique solution?

Given IVP, for $x \in (-5, 5) $ and $ t \in R $ $$ \frac{dx}{dt} = \sqrt{|x|}$$ $$ x(0) = 0 $$ I want to find if it has a unique solution or not. So I use Picard theorem: Either I check if $\sqrt(|...
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2answers
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Let $f : [0, 1] \to \mathbb{R}$ be a strictly increasing function and let $c\in (0, 1)$. If $\lim_{x \to c} f(x) = L < \infty$, prove that $f(c) = L$. [closed]

I Tried a contradiction, but couldn't see where to use the fact that the function is strictly increasing, I'm not sure if this is the correct approach. Any help would be appreciated.
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2answers
59 views

Continuity on an open interval

Question: Let $f : (0, 1) \to \mathbb R$ be continuous on $(0, 1)$ with $\lim_{x\to 0+} f(x) = 0 $ and $\lim_{x\to 1−} f(x) = 1$. Prove that if $\lambda \in (0, 1) $ then there exists $ c \in (0, 1) $ ...
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2answers
36 views

Prove the continuity of the map defined on locally compact Hausdorff space

A ternary Banach space $(A,[,])$ is complete normed space with a linear ternary map $[,]:A\times A\times A \to A$ satisfying $\vert \vert[a,b,c] \vert \vert \leq \vert \vert a \vert \vert \vert \vert ...
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1answer
67 views

Limit of a composite function that is not continuous

Below is a link to the graph that I'm working with. Graph of function f From this graph, based on my understanding: $$\lim_{x\rightarrow 2} f(x)$$ The limit does not exist. This is because as $x$ ...
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1answer
35 views

Given point $A$ in the interior of a circle and point $B$ outside the same circle, prove that there is a point in the circle in $\overline{AB}$

I want to run away from continuity/analysis and atempt to prove it using euclidean geometry theorems. I do know that the interior of a circle is a convex region. But I lack a smart way to prove the ...
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1answer
55 views

Give an example of continuously differentiable function that satisfy some properties

Can you give an example of continuously differentiable function that satisfy this properties: Let $\{(a_i, b_i)\}_{i=1}^{\infty}$ be the family of open subintervals of $[0,1]$ with rationals endpoints....
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17 views

A question about transition map on $L^1$-space

I have a question about the transition map $\tau_h u(x):=u(x+h)$ for $h\in\mathbb{R}^n$. Let say $u\in L^1(\mathbb{R}^n)$ and $\frac{\partial ^a}{\partial x^a}u\in L^1(\mathbb{R}^n)$ for every multi-...
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2answers
14 views

Convergence of functions in a certain point

Assume that $(f_n)$ is a sequence of continuous functions with $f_n:[0,1] \to \mathbb{R}$. Assume now that $(x_n)$ is a sequence of elements from the interval $[0,1]$ such that $x_n \to 0$ and $f_n(...
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2answers
33 views

Is it possible to disprove continuity using $\epsilon -\delta$ approach?

I am trying to prove that the function $$f(x)=\left\{\begin{array}{ll} 5 x-2 & \text { if } x \neq 1 \\ -2 & \text { if } x=1 \end{array}\right.$$ is Discontinuous at $x=1$ using $\epsilon -\...
6
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1answer
57 views

Operator targeting $\ell^2$-direct sum of Hilbert spaces continuous if all its projections are continuous?

Let $(U, \|\cdot\|_U)$ be a Banach space, $(H_k, \|\cdot\|_k)_{k\in\mathbb{N}}$ be a sequence of Hilbert spaces and denote by $$\tag{1}H:=\bigoplus_{k=1}^\infty H_k \equiv \left\{h=(h_k)\ \middle| \ ...
2
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0answers
26 views

How to interpret the categorical relationship between convergent sequences and continuous functions

I attend a math seminar at my university, and a senior professor went on the following (paraphrased) tangent As a side note, there's an interesting relationship between convergent sequences and ...
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2answers
48 views

Continuity of $\frac{x}{\sqrt{x^2+y^2}}$

I proved that the function $f(x,y) = \frac{x}{\sqrt{x^2+y^2}}$ at $x\neq0$ is continuous and want to extend continuity to $x=0$. Using polar coordinates I found that $\lim_{x,y\to 0} f(x,y)$ is ...
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0answers
18 views

uniform convergence on compacts in probability is preserved under continuous transformation

Suppose we have a sequence of $\mathbb{R}^d$-valued stochastic processes $(X^n)_{n \in \mathbb{N}}$ converging to $X$ uniformly on compacts in probability, i.e. $$ \operatorname{plim}_{n \to \infty} \...

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