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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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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How can I prove that this function is continuous in (0,0)?

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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31 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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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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24 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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25 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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55 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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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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21 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 ...
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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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32 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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32 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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224 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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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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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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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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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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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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68 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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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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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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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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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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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 ...
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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, $\mathcal{G}$ ...
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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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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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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 ...
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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 ...
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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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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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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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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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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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function of 2 variables [on hold]

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
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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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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} ...
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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 ...
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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 ...
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1answer
40 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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22 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 \\ ...
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1answer
16 views

Functional limits and definition of continuity - difference and implications?

Continuity: A function $f : A → \mathbb{R}$ is continuous at a point $c ∈ A$ if, for all $\epsilon > 0$, there exists a $δ > 0$ such that whenever $|x − c| < δ$ (and $x ∈ A$) it follows that ...
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1answer
34 views

Does $\Vert f-s_n \Vert_\infty \to 0$ still hold for $f\in C^0[a,b]$?

If $f\in C^2[a,b]$ and $s_n$ its piecewise linear interpolation at points $x_0, \ldots, x_n$ with $h_n = \max_{j=0,\ldots,n-1} (x_{j+1}-x_j)$ then one can show that $$\Vert f-s_n \Vert_\infty \leq ...
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3answers
28 views

$T$ continuous in $x_0$ then $T$ is continuous

Let $T:V\to W$ be a linear operator, with $V, W$ normed spaces. Show that if there exist $x_0 \in V$ such that $T$ is continuous in $x_0$ then $T$ is continuous. I'm thinking that given an ...
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1answer
47 views

Are bounded analytic functions on the unit disk continuous on the unit circle?

Let $f(z)$ be holomorphic on the open disk $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Moreover, let $f$ be bounded on the boundary of $\mathbb{D}$, i.e. $$ \sup_{\varphi \in [0,2\pi]} ...