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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3
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1answer
55 views

Prove that if $f$ is continuous and differentiable on $\mathbb{R}$ and has three roots, then its derivative $f′$ has at least two roots.

I know that we are supposed to use Mean Value THeorem for this question. So from the theorem, if $f$ is continuous on an interval $[a,b]$ and has two roots, this means that there is a point $c\in [a,b]...
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1answer
39 views

Prove continuity for an integral function [on hold]

Let $f$ a continuous real function defined on $[0,1]$ be given. It's asked to prove continuity in $x \in [0,1]$ for the function $$x \to \int_{0}^x \frac{f(t)}{(x-t)^{\frac{1}{2}}}dt$$ Thanks in ...
4
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2answers
77 views

Limits and Continuity in Multi variable calculus.

Checking continuity of $f(x,y)$ at $(0,0)$: $$ f(x,y)=\begin{cases}\dfrac{x^3+y^3}{x-y}\ \ ,x\neq y\\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ,x=y \end{cases}$$ Using polar coordinates $x=r\cos\theta$ and $...
1
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3answers
23 views

Investigating Continuity of Dirichlet and related functions: An $\epsilon-\delta$ approach

I have trouble proving discontinuity of the Dirichlet function, using the $\epsilon-\delta$ approach. The function is defined as follows: $$ f(x) = \left\{\begin{array}{l l} 1 &\text{if }x ...
2
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2answers
29 views

Finding appropriate $\delta$ to prove continuity

I want to show that $f(x) = \sqrt{x^2 + 5}$ is continuous at $x = \pi$ using the epsilon delta definition. Here's how far I could go: $|x - \pi| < \delta$ is same as $|x^2 - \pi^2| < \delta|x+\...
1
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0answers
23 views

Discontinuity of norm on $\ell^p$ defined by vector-space isomorphism to $\ell^q$

I'm reading the paper by Dijkstra and van Mill called Topological equivalence of discontinuous norms, and in its introduction there is this: Consider the case of $\ell^p$ and $\ell^q$, where $p<...
0
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1answer
24 views

$g(z)=z^2~$ What will $~g(y ̅_n )~$ converge to in probability?

Another homework question here. $~Y_1, Y_2, \cdots, Y_n~$ is i.i.d. $$g(z)=z^2$$ What will $~g(y ̅_n )~$ converge to in probability? I'm not sure if Slutsky Theorem has something to do with it. ...
0
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2answers
25 views

Doubt in Proof of Boundedness for Continuous Functions

By now, it has been established that the image of a continuous function over a closed bounded interval is also bounded. The proof given below aims to show that the maximum value is attained at some ...
0
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0answers
5 views

(In Editing) Compactness of graph on compact domain can prove the continuity of the function?

I do not understand the proof of Exercise 4.6 of Baby Rudin written by Roger Cooke(https://minds.wisconsin.edu/bitstream/handle/1793/67009/rudin%20ch%204.pdf?sequence=8&isAllowed=y). Can someone ...
0
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1answer
55 views

Show function is continuous for all x and y [duplicate]

$$ \begin{cases}xy^2\sin\left(\frac 1 y\right), \text {if $y$ $\neq$ 0} \\ 0, \text{if $y$ = 0} \end{cases}$$ I'm not sure how to work out this limit as $x$ goes to infinity. If it were just the y ...
4
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0answers
66 views

$f(0)^2+f'(0)^2=4$ then $f(x_0)+f''(x_0)=0$ [duplicate]

If $f(x)$ is a real valued $C^2$ function on the real line such that its value is between $-1$ and $1$. Then, if $f(0)^2+f'(0)^2=4$, I have to show that there exists $x_0$ on the real line such that $...
0
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0answers
30 views

Continuity of a max function [duplicate]

I have a question regarding the continuity of a function. The problem states: let $f(x)$ be a continuous function for $a \leq x$ and define $h(x) = \max_{a\leq t \leq x}f(t)$ Prove or disprove ...
0
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2answers
29 views

Show that $f(x,y)$ is discontinuous at the point $(1,2)$

Let $f(x,y)=\begin{cases} & 2xy, \text{ if }(x,y) \neq (1,2) \\ & 0, \text{if } (x,y)=(1,2) \end{cases}$ I have to prove that the function is discontinuous at $(1,2)$. I can see from ...
1
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1answer
26 views

Will the inverse mapping also be continuous? [duplicate]

Suppose a map $f:A\to B$ is continuous and invertible. Will the inverse map $f^{-1}:B\to A$ be always continuous also?
1
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1answer
40 views

For cadlag functions $x,y$, if for each $t$ $x(t)=y(t)$ or $x(t)=y(t-)$ then $x=y$.

Let $x,y$ be cadlag functions on $[0,\infty)$. Suppose for each $t$, either $x(t)=y(t)$ or $x(t)=y(t-)$, then how does this imply that $x=y$? I am not sure how to show that for the case $x(t)=y(t-)$, ...
0
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1answer
43 views

Unique continuous function with…

I am not sure how to proceed on the following problem: Prove that there is a unique continuous fuction $f:[0,1]\to \mathbb{R}$, with the property that $f(x)=x+\int_0^1 \sin(2\pi (x-y))^2 f(y)dy$ for ...
0
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0answers
18 views

Continuity and Sobolev Spaces in an example on the direct method

Working on counterexamples to the direct method if a condition is missing I have the following problems: $1. F:W^{1,2}([0,1]) \rightarrow \mathbb{R_\infty}, F(u)= \int_{0}^{1} (x*u(x))^2 dx$ if $u(0)=...
3
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1answer
35 views

Example of a function which is continuous in $\mathbb{R}_K$ but not in $\mathbb{R}_u$

Let $\mathbb{R}_u$ be $\mathbb{R}$ equipped with the usual topology and $\mathbb{R}_K$ be $\mathbb{R}$ equipped with the $K$-topology. Is there a function $f : \mathbb{R} \to \mathbb{R}$ which is ...
5
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2answers
128 views

Are there $f:\mathbb{R}^{+}\times\mathbb R^+\rightarrow\mathbb{R}^{+}$ associative and not conjugate to the addition?

We say that Definition A function $f:X\times X\rightarrow X$ is associative if $\forall a,b,c \in X\; f(f(a,b),c)=f(a,f(b,c))$ Definition Associative functions $f,g:X\times X\rightarrow X$ ...
2
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0answers
32 views

Multiplicative maps on C*

Suppose $\mathbb{C}$ is the complex plane and $\mathbb{C}^{*} = \mathbb{C} \setminus \lbrace 0 \rbrace$. Let $ \varphi : \mathbb{C}^{*} \rightarrow \mathbb{C}^{*} $ be a continuous multiplicative ...
4
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1answer
43 views

If f(x_n) converges whenever (x_n), then f is continuous.

This is the original question: Let $f: X\to Y$ be a map between metric spaces. Prove that if $(f(x_n))_{n=0}^\infty$ converges in $Y$ whenever $(x_n)_{n=0}^\infty$ converges in $X$ then $f$ ...
0
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0answers
36 views

Expressing a continuous function in terms of a fixed function plus an arbitrary function

In calculus of variation, they would often claim that any well-behaved function $\tilde{y}(x)$ can be expressed as $y(x)+\epsilon g(x)$, where $y(x)$ is our stationary function. I cannot find the ...
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0answers
18 views

Is positive part Lipschitz continuous?

Is the positive part of a convex function Lipschitz continuous if the positive part of function $f(x)$ is defined as $\max\{f(x),0\}$.
5
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5answers
71 views

Proving the Continuity of $e^x$

Can one say that $e^x$ is the sum of an infinite number of terms (Taylor expansion), every term being a continuous polynomial in itself, the sum of all the terms is continuous and so $e^x$ is ...
5
votes
2answers
91 views

Continuity of $x\sin\frac{1}{y}$ at $(x, 0)$

I need to check if function is continue in $(x,0)$ $$f(x,y) = \left\{ \begin{array}{ll} x\sin\frac{1}{y} & \mbox{if } y \ne 0 \\ 0 & \mbox{if } y = 0 \end{array} \right.$$ Can someone ...
0
votes
3answers
49 views

Prove that exist $c \in (a;b)$ such that $1975 f(a)+297 f(b)=2004 f(c)$

Let $f(x)$ be a polynomial, $a<b$ such that $f(a)\neq f(b)$. Prove that exists $c\in (a;b)$ such that $1975 f(a)+297 f(b)=2004 f(c)$. Please help me! Thanks so much!
1
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0answers
21 views

Is this proof correct? (I'm tryng to use the continuity of a inverse matrix and its determinant)

I have a vector function $F:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^n$, depending upon the variables $(\beta,x_1,\dots,x_n)$ and it is continuously differentiable for any $(\beta,x_1,\dots,x_n)\in\Omega\...
0
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1answer
15 views

How the solve the following problem regarding additivity of the function $f$ [duplicate]

I am trying to solve the problem. How we can use the continuity concept?
4
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1answer
126 views

Limit of composite function at isolated points

Paul's Online Notes (link) writes: I have some doubts about the above claim. In particular, I'm not sure if it's correct if $b$ is an isolated point of the domain of $f$. I have tried to come up ...
0
votes
2answers
22 views

Continuity at a point and open preimage

Let $(X, d_X)$ and $(Y, d_Y)$ be two metric spaces, let $f:X \mapsto Y$ be continuous at a point $x_0$, that is: for any neighborhood $V$ of $f(x_0)$ there exists an open set $U$ such that: $x_0 \in U ...
1
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1answer
41 views

Uniformly continuous functions in terms of limits

In terms of limits, a function is continuous at a point $a$, if $\lim_{x\to a} f(x) = f(a)$. Now, what can we say about uniformly continuous functions in terms of limits?
5
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1answer
80 views

How to show that this function is continuous?

Let $\psi(x,y)$ be a continuous function in two real variables, and then define $$f(x) = \sup_{t \in [-x,x]} \psi(x,t)$$ It seems to me that the function $f$ should also be continuous for $x \geq 0$, ...
1
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0answers
27 views

Definition of continuous representation of topological group

I have a question about the definition of continuous representation of topological group. Let $G$ be a group and $\rho : G \rightarrow {\rm Aut}_k(V)$ be a representation of $G$ , where $k$ is a ...
2
votes
1answer
79 views

Is there a continuous function $f\colon \mathbb{R} \to \mathbb{R}$ such that $f(\mathbb{Q}) \subseteq \mathbb{R} \setminus \mathbb{Q}$?

Is there a continuous function $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that $f(\mathbb{Q}) \subseteq \mathbb{R} \setminus \mathbb{Q}$ and $ f(\mathbb{R} \setminus \mathbb{Q}) \subseteq \...
0
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1answer
58 views

Differentiable manifold continuous with respect to parameters?

Let $h : U \times \mathbb{R}^n \to \mathbb{R}^m$ be a continuously differentiable function, where $U$ is an open of $\mathbb{R}^p$, to be understood as the space of parameters. Assume that for all $(u,...
1
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1answer
40 views

Definition of continuity at a point: can we take only sequences of distinct members?

Q. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function. Fix $a\in \mathbb{R}$. Suppose $f$ satisfies property: given any sequence $\{x_n\}$ in $\mathbb{R}$ of distinct terms converging to $a$, $f(...
0
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3answers
26 views

Is $f:\mathbb R^3 \to \mathbb R$, $f(x,y,z)=x+e^yz$ Lipschitz?

Define $f:\mathbb R^3\to \mathbb R$ as $f(x,y,z)=x+e^yz$. Is $f$ Lipschitz? I'm having a hard time with this question. Simply chugging $(x,y,z),(x',y',z') \in \mathbb R^3$ and calculating $$|f(x,y,z)-...
0
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0answers
26 views

bounding the difference between two function compositions

I have two smooth functions, $a$, $b$, on a compact domain $[0,1]$ into the reals, and I want to know if the following bound is true for all $x\in [0,1]$: $|a\circ f(x) - b\circ g(x)|\leq|a - b|_\...
0
votes
1answer
36 views

Prove a matrix function is continuous

Suppose $f:(C[0,2],\lVert\rVert_{\infty})\to (M_{2}(\mathbb{R}), \lVert T\rVert_{op})$, where $\lVert T\rVert_{op} = \sup_{\lVert \mathbf{x}\rVert =1}\lVert T\mathbf{x}\rVert$ Let $A=U^{-1}DU$ be ...
-1
votes
1answer
66 views

Let $f(x) : [0,1] \to \Bbb R$ be continuous function. Prove that there exist $x \in [0, 1]$ such that $f(x) = x$

Let $f(x) : [0,1] \to \Bbb R$ be continuous function. Prove that there exist $x \in [0, 1]$ such that $f(x) = x$. I couldn't find any possible duplicate for this question. If you found please feel ...
0
votes
2answers
55 views

Can a continuous function send bounded domain to unbouded domain?

Let $f:D=\overline{D(a,r)}\subset \mathbb{C} \to \mathbb{C}$ be a continuous function whose image of $D'=\{z \ | \ |z-a|=r\}$ is homeomorphic to circle $S^1$. Then I guess the image of $f$ should be ...
0
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0answers
37 views

Is there any function continuous only at rationals? [duplicate]

If I define a constant function on domain of rationals, then is it the right chosen function to be continuous at rationals? Also is there any function with Domain of reals which is continuous on ...
0
votes
0answers
21 views

Isit possible there exists a change of variable that affects Clairaut's theorem

I have that question, if I define a change of variable that might change the continuity of a function (I am not sure if that is even possible), or the continuity of the partial derivatives, it is ...
1
vote
0answers
37 views

Prove$f(x)=\int_{2}^{x^2}\int_{0}^{s}e^{-t^2}\ dtds$ is continuous and convex

I want to show: $$f(x)=\int_{2}^{x^2}\int_{0}^{s}e^{-t^2}\ dtds$$ is a continuous and convex function Consider: $h(s)=\int_{0}^{s}e^{-t^2}\ dt$ Then I get: $f(x)=\int_{2}^{x^2}h(s)\ ds$ ...
6
votes
0answers
60 views

Continuity at the origin of $f(x,y)$ given by $0$ there, and $\frac{\sqrt{|x||y|}}{|x|+|y|}$ elsewhere

Define the function $f:\mathbb{R}^2\mapsto \mathbb{R}$ by \begin{equation*} f(x,y) := \begin{cases} \frac{\sqrt{|x||y|}}{|x|+|y|} & \textit{if} \; \; (x,y)\neq (0,0) \\ 0 & \textit{if} \; \; ...
4
votes
1answer
31 views

Prove that a ball with radius $r<1$ is a subset of the range of approximate identity function from a unit ball

I am struggling with finding a proof (or counterexample) for the following Theorem: Let $x\in\mathbb{R}^n$ and $f:\mathbb{R}^n\to \mathbb{R}^n$ a continuous function. Assume $\forall x\in B_1$, $||x-...
0
votes
1answer
55 views

Proving continuity of the following piecewise function

Consider the following function : $$f(x) = \begin{cases} x^2-9 \ & \text{if}\ x \le 4 \\[2ex] \frac{2x^2-9x+4}{x-4} & \text{if}\ x >4 \end{cases}$$ Now I want to show that this function ...
0
votes
0answers
26 views

Example of a function such that $A$ and $\alpha$ depend on $\epsilon$ from Spivak's Calculus

The following are two conditions needed for a proof below: $(i)$ if $x$ and $y$ are in $[a,b]$ and $|x-y|<\delta_{1}$, then $|f(x)-f(y)|<\epsilon$, $(ii)$ if $x$ and $y$ are in $[b,c]$ ...
0
votes
1answer
29 views

metric on infinite product, continuous

Let $(X_k,d_k)_{k\in\mathbb{N}}$ be a family of metric spaces. Let $X=\prod_{k=1}^\infty X_k$ be equipped with the metric $$d(u,v)=\sum_{k=1}^\infty 2^{-k}\overline{d_k}(u_k,v_k),$$ where $\...
0
votes
0answers
24 views

The sum of two functions Holder continuous is Holder continuous?

Suppose that $\varphi = \phi - \psi$, where $\varphi, \phi, \psi: M \rightarrow\mathbb{R}$ and $M$ is a compact metric space. If $\beta_1, \beta_2, \alpha_1, \alpha_2$ are positive constants fixed ...