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) ...

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Upper semi-continuity results

I have recently been introduced to the notion of upper semi-continuity on a metric space $X$. Please advise on the following queries: If $f:X \rightarrow \mathbb{R}$ is upper semi-continuous and ...
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2answers
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Is my proof for this limit correct?

I want to prove that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to 2. Let $a_0$ = $\sqrt{2}$ $a_n$= $\sqrt{2+a_{n-1}}$. Then, proving that $\sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}$ limits to ...
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1answer
10 views

Limiting and continuous about one function

I have a function which is \begin{equation} F(x)= \begin{cases} f(x) & x \in [\underline{x},\bar{x})\\ \\ f(\bar{x}) & x=\bar{x} \end{cases} \end{equation} The function $f(x)$ is strictly ...
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1answer
31 views

which hypothesis for boundedness of this function

Let $v:[0,\infty)\rightarrow \mathbb{R}_+$ be a positive function such that $$\exists T,q>0\,\,s.t.\,\, \forall t\in[0,\infty),\,\,\int_t^{t+T} v(\tau) d\tau \le q$$ I'm looking for the "less ...
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1answer
44 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
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1answer
48 views

How can I prove that this function is continuous in (0,0)? [on hold]

I have this function: $$ \lim_{(x,y)\to (0,0)} = \frac{2(1-\cos(xy))+\arctan(x^4)-x^2(x^2+y^2)}{(x^2+y^2)^\alpha} $$ I have to find which $ \alpha$ makes the function continuous. But my first problem ...
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1answer
33 views

Positive derivative on [0,1] implies a continuous derivative on [0,1]

If a real-valued function F defined on [0,1] is differentiable with positive derivative f everywhere on [0,1], can we conclude that f is continuous?
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1answer
56 views

Alternative Proof of the Extreme Value Theorem

I have proven the Boundedness Theorem for continuous functions and would now like to prove the Extreme Value Theorem; that is, show that the upper bound is indeed attained for continuous functions. I ...
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1answer
26 views

Distance of a point to a subset.

Let $(M,d)$ be a metric space. For a subset $A\subseteq M$ we define the distance of a point $x$ to $A$ as $$\alpha_A(x):=\operatorname{dist}(x,A):=\inf_{y\in A}d(x,y)$$ Prove that: ...
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1answer
28 views

Continuity of composite functions

The continuity theorem for composite functions states that if $f(x)$ is continuous at $x = a$ and $g(x)$ is continuous at $x = a$ , then the composite function $f\circ g$ and $g\circ f$ are also ...
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57 views

Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
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2answers
18 views

Is homeomorphic image of closed bounded subsets of metric spaces , also closed bounded in the homeomorphic image metric space?

Let $X$ , $Y$ be homeomorphic metric spaces with homeomorphism $f$ , then is it true that for any closed bounded subset $A$ of $X$ , $f(A)$ is also closed and bounded in $Y$ ?
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0answers
22 views

If for every $a > 0$, $u \in C^\infty([a,\infty))$, then is $u \in C^\infty((0,\infty))$?

Suppose that for every $a > 0$, $u \in C^\infty([a,\infty))$. Does this imply that $u \in C^\infty((0,\infty))$? I think it is true when we just work in $C^0$, but with $C^\infty$ you need to ...
2
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0answers
26 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
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0answers
33 views

homeomorphism as a result of other homeomorphisms

If $$B = \bigcup_{R>0} B_R$$ and all the identities $$\operatorname{id}_R : (B_R,d_1) \rightarrow (B_R,d_2)$$ for $R>0$ are homeomorphisms, then is $$ \operatorname{id} : (B,d_1) \rightarrow ...
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1answer
34 views

finding and proving where function is…

So I have this function: $ f(x) = \begin{cases} ( 2 \sqrt{-1-x}-1)^{\frac{1}{4^{-x}-16}} & \quad \text{if } x<{-2}\\ - \frac{\pi}{4}x & \quad \text{if } -2\leq x \leq 1 \\ \frac{\sin{(\pi ...
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1answer
38 views

Intuition on the Topological definition of continuity, considering the special case of the step function.

I'm trying to get an intuition for open sets and topological reasoning in general. One example I want to understand is the step function, and specifically why it would be considered discontinuous ...
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3answers
225 views

Solve this functional equation:

Functional equations such as this one appear only once every several years on exams, so I feel it's hard to have a sure-fire way to approach the problem, unlike, say, solving a series convergence ...
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0answers
30 views

Relation between $\lim_{a \to 0}\int_a^T u(t)$ and the Lebesgue integral $\int_0^T u(t)$

Let $u\colon (0,T] \to \mathbb{R}$ be function with $u \geq 0$ everywhere and $u$ is continuous on $[a,T]$ for every $a > 0$. Suppose that the limit $$\lim_{a \to 0}\int_a^T u(t) \;dt ...
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1answer
38 views

When can I take $\lim_{a \to 0}\int_a^T u$?

Suppose I have a function $u:(0,T) \to \mathbb{R}$ which is integrable over $[a,T]$ for every $a > 0$, and I have the results $$\int_a^T u = U(T)-U(a)$$ for such $a$. When am I allowed to conclude ...
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1answer
25 views

Help with vacuous continuous function (please)

i have a question that's been bugging me for the past two days. The definition of a function that is continuous at some point $a$ of it's domain, states: $f$ is continuous at $a$ if $$\lim_{x\to a} ...
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1answer
27 views

Strictly monotonic increasing function with a closed domain and range

Let $a,b,c,d \in \mathbb{R}$ with $a<b$, $I = [a,b]$. Let $f: I \rightarrow \mathbb{R}$ be a monotonic, strictly increasing function. Also $c<d$ and $f([a,b]) =[c,d]$ a) Proof that $f$ is ...
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1answer
30 views

The image of the inverse of a continuous function

First of all I'm not sure if my title is correct with the question, I find it hard to really get about what kind of set this question is about. It would be very helpful if someone could explain this ...
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0answers
17 views

Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
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3answers
59 views

prove continuity

Let $ f:\Bbb R \to \Bbb R $ satisfy the property $ f(x+y)=f(x)+f(y)$ for all $x,y$ in $ \Bbb R $ I have to show that 1)$f(0)=0 , f(-x)=-f(x),$ for all $x$ in $\Bbb R$, and $f(x-y)=f(x)-f(y)$ $y$ in ...
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3answers
69 views

Show that $f$ is continuous mathematically.

Let $f:[0,\infty)\to \mathbb{R}$ be given by $f(x)=\sqrt{x}$. Show that it is continuous. This is taken from Example 3.7 on <link> page 22 on the paper. It has shown that it is continuous at ...
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0answers
26 views

Is a function $f \in \mathbb{C}^{ \infty}[0,l]$ always in $L^2(0,l)$?

I was trying to find a function that is not in $L^2(0,l)$ but that it is in $\mathbb{C}^{\infty}[0,l]$ for l>0. But if the function is continuous at both sides of the interval then it is integrable, ...
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0answers
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Is there a function Lipschitz on the right of every point, but everywhere discontinuous?

Today I came across the following definition: Definition: A function $f:[a,b] \to \Bbb C$ is Lipschitz to the right of $t_0 \in [a,b]$ if exists $L>0$ such that $|f(s+t_0)-f(t_0^+)| <Ls$ for ...
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1answer
28 views

Upper semi-continuity proof for topological spaces

Hi does anyone have any idea or a possible hint for a proof of the following result: Consider asymmetric norm $p$ on $\mathbb{R}$ given by $p(t) = t^{+}$, for $t \in \mathbb{R}$. Show that if $(X, ...
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2answers
56 views

Is this Function differentiable and continuous at x=0? [closed]

Is $f(x)$ continuous and differentiable at $x = 0$ ? $$f(x) = x(\sqrt{x} - \sqrt{x+1})$$
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2answers
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If you have a function $f(x)=\frac{x^2}{x}$, then is the function continuous at x=0?

If you have a function $f(x)=\dfrac{x^2}{x}$, then is the function continuous at $x=0$? On one hand, if you simplify it and end up with $f(x)=x$, it is continuous at $0$, but if you keep it in its ...
2
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1answer
52 views

Anywhere I integrate $f_n$, the integral approaches $f$. Is $\lim_n f_n = f$ a.e.?

Something tells me this is obvious... I have a bunch of functions: $f,f_n:\mathbb{R}^2\rightarrow \mathbb{R}$, all integrable. Also, $f$ is continuous. I also have a family of sets, $\mathcal{G}$ ...
2
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1answer
45 views

Every continuous map of a closed interval into itself has a fixed point

The Question: Please show this theorem: Let $f: I=[a,b] \rightarrow \mathbb{R}$ be a continuous map such that $f(I) \supset I $. Then $f$ has a fixed point on I. My Attempt: Suppose there is a ...
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3answers
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Looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points

I am looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points ; please help , thanks in advance .
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1answer
20 views

Is this statement equivalent to $f(x)\in\mathscr C(a,b)$?

I'm pondering on the following: $$f(x)\in\mathscr C(a,b)\overset{?}{\Longleftrightarrow} f(x)\in\mathscr C[a+\delta,b-\delta]\quad\forall\delta\in(0,\frac12(b-a)) $$ I believe it's true. The ...
6
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2answers
48 views

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ?

Let $X \subseteq \mathbb R$ and $X$ has same cardinality as $\mathbb R$ , does there always exist a continuous surjection from $\mathbb R$ onto $X$ ? ( I know that there need not always be a ...
11
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1answer
171 views

To find continuous functions on $\mathbb R$ which preserve certain algebraic structures

Can we determine all non-constant continuous functions $f:\mathbb R \to \mathbb R$ such that for every subgroup $G$ of $(\mathbb R,+)$, $f(G)$ is also a subgroup of $(\mathbb R,+) $ ? And ...
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45 views

Finding all continuous $f$ on $\mathbb R$ such that for each $r\in\Bbb R\setminus\Bbb Q $ , $f(rx)/f(x)$ is constant $\forall x\ne 0$?

Can we determine all continuous functions $f:\mathbb R \to \mathbb R$ such that for every $r \in \mathbb R \setminus \mathbb Q$ , $\exists k_r \in \mathbb R$ such that $f(rx)=k_rf(x) , \forall x \in ...
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1answer
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Convergence of the image of a sequence in topological sense

I haven't been able to find it, but i'm sure this question has been answered since it is a fundamemtal one: Let $(X, \tau $) and $(Y, \tau $) be topological spaces. Let $f: X\to Y $ be continuous. If ...
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1answer
53 views

Methods to prove that a function is continuous

Although I seem to understand the concept of continuity in connection with functions, I am often stuck proving that particular functions are continuous. I think the epsilon-delta definition is the ...
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2answers
35 views

Find where $f$ is continuous

We have a function $f: \mathbb{R} \to \mathbb{R}$ defined as $$\begin{cases} x; \ \ x \notin \mathbb{Q} \\ \frac{m}{2n+1}; \ \ x=\frac{m}{n}, m\in \mathbb{Z}, n \in \mathbb{N} \ \ \ \text{$m$ and $n$ ...
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3answers
41 views

Continuous functions

I have a question for you. Let $f\colon\mathbb{R}\rightarrow \mathbb{R}$ continuous. Assume that there exists $s,t\in\mathbb{R}$, with $t>s$, such that $f(s)=0$ and $f(t)>0$. I want to prove ...
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1answer
39 views

Continuous function approximated by a polynomial

I have to prove that: If $f$ is a real valued continuous function on the closed interval $[a,b]$ then given $\varepsilon>0$ there is a polynomial $p(x)$ such that $p(a)=f(a)$, $p'(a)= 0$ ...
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1answer
39 views

function of 2 variables [closed]

we have the next function: $$f(x,y)=\begin{cases} \dfrac{\sqrt{x^2y^2+1}-1-x^2-y^2}{x^2+y^2} & (x,y)\neq (0,0) \\ c & (x,y)=(0,0) \end{cases}$$ Is there $c$ that $f(x, y)$ is continuous ...
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1answer
21 views

Metric spaces - continuity - open/closed.

Let $f:(M_1,d_1)\to (M_2,d_2)$ be a mapping between two metric spaces. a)Let $A\subseteq M_1$ be open and $B\subseteq M_1$ closed. Show through the use of counterexamples that in general ...
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20 views

continuity of linear functional on family of functions

If $A$ : $C[a,b]\rightarrow \mathbb{R}$ is a continuous linear functional, then $ t\mapsto A(f_{t})$ is a continuous function on $\mathbb{R}$. where \begin{align} f_t(x)= \left\{ \begin{array}{lr} ...
3
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1answer
67 views

If $f_n \to f$ uniformly and $f$ is continuous, does that imply $f_n$ is continuous?

I have a theorem in my book which says if $(f_n)$ is a sequence of functions uniformly converging on $A$ to $f$, and is continuous at some point $c \in A$, then $f$ is also continuous at this point ...
2
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0answers
21 views

Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
3
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1answer
41 views

Continuity of the locus of the maximum of a two variables real function

Suppose that $$\begin{array}{lrcl} f : & [0,1]^2 & \longrightarrow & \mathbb{R} \\ & (x,y) & \longmapsto & f(x,y) \end{array}$$ is a continuous function and that for all $x ...
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0answers
23 views

piecewise defined function finding at which points it is continuous

We have: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ $$ f(x) = \begin{cases} x^3 - 3x + 2 &\text{if }x \in \mathbb{Q} \\ x^3 + x^2 + 4 & \text{if }x \in\mathbb R\setminus\mathbb Q \\ ...