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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Calculus: Limit and continuity

Would anyone mind telling me how to solve these two questions? I know it sounds silly but I really have no idea.
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epsilon/delta definition alternative?

The phrase "any epsilon greater than zero" has always seemed somewhat vague. Question: is this an equivalent definition? $\forall\mbox{ Natural Numbers }N>0 \,\,\exists\delta>0\mbox{ s.t. ...
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270 views

$f(f(\sqrt{2}))=\sqrt{2}$ then f has a fixed point

$f(x)$ is continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ $f(f(\sqrt{2}))=\sqrt{2}$ Prove that $f$ has a fixed point in other words prove the there is $x_1$ such that $f(x_1)=x_1$ I tried using ...
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$f \in \mathcal{R}(\alpha)$ on $[a,b]$, then $\exists P_n$ s.t. $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$.

Assume $f \in \mathcal{R}(\alpha)$ on $[a,b]$, and prove that there are polynomial $P_n$ such that $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$. This is what I have, ...
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Continuity from complete metric space

Let $f:X\rightarrow Y$ be a continuous function, such as: $f(X)=Y$. If $X$ is complete, does it imply $Y$ is complete?
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57 views

Differentiability with non continuous partials (origin)

The function $$f(x,y)= \frac{x^{2}y^{2}}{(x^{2}+y^{4})} \quad if \quad (x,y) \neq (0,0)$$ $$f(0,0)=0$$ In order to study it's differenciability at the origin, I've studied if the partial are ...
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$\sqrt{x}$ isn't Lipschitz function

A function f such that $$ |f(x)-f(y)| \leq C|x-y| $$ for all $x$ and $y$, where $C$ is a constant independent of $x$ and $y$, is called a Lipschitz function show that $f(x)=\sqrt{x}\hspace{3mm} ...
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Continuity and differentiability of $x^a\sin ({1\over x}) $ at $0$

Consider the function $$ g_a (x) = \begin{cases} x^a\sin ({1\over x}) & x \neq 0 \\ 0 & x=0 \end{cases}$$ I am looking to determine for which $a$ the map $g_a$ is differentiable on ...
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170 views

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...
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let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume $f(z) = f(2z)$, prove that f is constant

$f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function and assume that $f(z) = f(2z)$ for all $z \in \mathbb{C}$. Prove that f is constant... Then we are supposed to use this result to ...
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282 views

$\{f_n\}$ equicontinuous sequence of functions on compact $K$, converges pointwise on $K$ then converges uniformly on $K$.

Suppose $\{f_n\}$ is an equicontinuous sequence of functions on a compact set $K$ and $\{f_n\}_{n=1}^{\infty}$ converges pointwise on $K$. Prove that $\{f_n\}_{n=1}^{\infty}$ converges uniformly on ...
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37 views

Continuous function and its extrema

Let $f(x)$ be a continuous function in $(a,b)$ and it has in this interval $m$ local maximum points and $n$ local minimum points. Then : $|m-n|\leq1$ It seems very obvious but is there any simple ...
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182 views

Rolle theorem on infinite interval

We have : f(x) is continues in $[1,\infty]$ and differentiable in $(1,\infty)$ $\lim_{x \to \infty}f(x) = f(1)$ we have to $\textbf{prove}$ that : there is $b\in(1,\infty)$ such that $f'(b) = 0$ ...
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Rationals, irrationals and Continuity

Suppose we have a continuous real valued function $f(x)$ which takes the form of a polynomial for all rationals, i.e. $\exists$ $a_0, a_1, ..., a_n \in \mathbb R$ such that $$f(x)=a_0x^n+a_1x^{n-1}+ ...
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Bourbaki on the fact that continuous function on a compact is uniformly continuous

I am now looking theorem 2 in paragraph 4.1 of: Bourbaki. "Elements of Mathematics General Topology. Part 1". THEOREM 2. Every continuous mapping $f$ of a compact space $X$ into a uniform space $X'$ ...
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101 views

Prove that a mapping $f:[-1,1]^2\to\mathbb R^2$ with certain properties has the value $(0,0)$.

The mapping $f:[-1,1]^2\to\mathbb R^2$ is known to be continuous. Also the image of the upper edge of the rectangle is contained in the upper half-plane, the left edge's image is contained in the left ...
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1answer
32 views

Proving Continuity in Multiple Variables

The Exercise: $f(x,y)=xy/(x^2+y^2)$ if $x \ne 0$ $f(x,y)=0$ otherwise Where is $f$ continuous? My Attempt: At $(0,0)$, let y=kx. $lim_{(x,y)\to (0,0)}f(x,y) = lim_{x\to 0}f(x,kx)=k/(1+k^2)$ which ...
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937 views

integral from zero to zero

it seems obvious that this integral is zero and so is the limit but what theorem we are using here? I see it's connected to Riemann sums with an interval=zero Right ? The function $\mathrm{f}$ is ...
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1answer
34 views

Continuity of monotone increasing function

Let f(x) be monotone increasing. Define $S=\{x|c \leq f(x) \leq d\}$. Show that S must be a single interval. I understand this intuitively and graphically, but don't know how to prove this formally. ...
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27 views

Continuity on finite interval

Problem: $f$ is a continuous function on $[a,b]$, and $f(x)>0$ for all $x$ in $[a,b]$. Prove that there is an $\alpha>0$ such that $f(x)>\alpha$ for all $x$. Please help in the steps of this ...
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46 views

Weak Lower Semicontinuity Generalized to any $L_{p}$ space

I am having difficulty with the following proof: Generalize the weak lower semi-continuity of$L^{p}$ norms to all $1\leq p < \infty$; i.e., show that if $u_{n}\to u$ weakly in $L^{p}$, then ...
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$C^0(\overline{\Omega}) \subset L^{\infty}(\Omega) ?$

In my functional analysis script there is an example that uses $$C^0(\overline{\Omega}) \subset L^{\infty}(\Omega) $$ where $\Omega \subset \mathbb{R}^n$ is an open subset and we take $L^{\infty}$ ...
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46 views

$f$ a continuous function, $f^{1/n}$ converges uniformly. How many zeros of $f$?

Let $f$ be a non-negative continuous function on the interval [0,1]. Suppose that the sequence $f^{1/n}$ converges uniformly. How many zeros does $f$ have? I'm confused about what this question ...
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89 views

Show that the function $ f(x)=\frac{1}{x+1}$ is continuous at $x=1$ using the $\varepsilon$-$\delta$ definition of continuity

Show that the function $ f(x)=\frac{1}{x+1}$ is continuous at $x=1$ using the $\varepsilon$-$\delta$ definition of continuity. My initial thoughts were that this is a straight plug in of value at ...
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2answers
99 views

Compact Domain and Inverse Image

I am trying to show that given $f:M \rightarrow N$, where $M$ is compact, $f$ is continuous and onto, then given $A \subset N$: $$ f^{-1}(A) \text{ closed} \implies A\text{ closed} $$ I am dealing ...
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Continuity formal proof

I have a problem similar to the one answers in this post (Mhenni Benghorbal's answer) $f(x)=x^2$, find $\delta$ such that $|x-1|\leq \delta$ implies $|f(x)-1|\leq \epsilon$. I am copying part of ...
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126 views

Monotone convergence of continuous functions

Is it true that for a sequence of functions $f_n \in C(K,\mathbb{R})$, where $K \subset \mathbb{R}^n$ the limit is semicontinuous? I would say that if it's a monotone decreasing lsequence, then the ...
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73 views

integral = zero

Let $f$ be a continuous function in $[a,b]$ For every continuous function $g(x)$ : $\displaystyle\int_{a}^{b}f(x)\cdot g(x)\;\mathrm dx=0$ We need to prove that $f(x)=0$. I thought about proving by ...
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27 views

Continuity under different metrics

On the real line R define the standard Euclidean metric d(x,y)=|x-y| and e(x,y)=|arctan(x)-arctan(y)|. Show that: (i) A function f:R→R is continuous under d if and only if it is continuous under e;
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132 views

Uniformly continuous function approximated by Lipschitz

Let $f~:~\mathbb R \to \mathbb R$ a uniformly continuous function. How could I construct a sequence $(f_n)_{n\ge 0}$ of Lipschitz functions such that $(f_n)$ uniformly converges towards $f$?
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Counterexample - Increasing function

My intuition says that the statement is false. Anyone out there know of counterexamples? Suppose $f: R\to R$ and $c\in R$ such that $f'(c) > 0$. So, $\exists \varepsilon> 0 $ such that ...
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285 views

Compactness implies Continuity?

I am stuck on this question (probably there are many counterexamples, but I can't find any). "Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ that preserves compactness (i.e, for every $K \subseteq R$, then ...
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Extending a holomorphic function

Let $D \subset \mathbb{C}$ be an open disc. Is there a function $$ f \in {\mathcal H}(\mathop D ) \cap C(\overline{D}), $$ such that, $\,\,f \notin {\mathcal H}(V)$, for every open set $V \supset ...
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205 views

Is $\cos\sqrt{xy}$ uniformly continuous?

I'm trying to find out if $$ f(x,y)=\cos\left(\sqrt{xy}\right) $$ is uniformly continuous on the set $\{(x, y)\in\mathbb{R}^2 : x\geq0, y\geq 0\}$. The theorems I have available to use for this are ...
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Two-variable limit, quotient of polynomials

I'm trying to evaluate the following limit, $$ \lim_{(x,y)\to(0,0)} \frac{x^3-y^2}{x^2-y} $$ which I think it doesn't exist, since for the curve $\alpha :[0,1]\to \mathbb R^2$, $\alpha(t) = (t, t^2)$ ...
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1answer
45 views

Continuosly differentation on composite functions

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ a $C^1$ function and defined $g(x) = f(\|x\|)$. Prove $g$ is $C^1$ on $\mathbb{R}^n\setminus\{0\}$. Give an example of $f$ such that $g$ is $C^1$ at the origin ...
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1answer
117 views

Hausdorff spaces

Let $X$, $Y$ be topological spaces. Assume that $X$ is a Hausdorff space, $D\subset X$ dense in $X$ and $f:X\to Y$ a continuous function. If $f$ when restricted to $D$ is a homeomorphism between $D$ ...
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show a function is continuous at irrational point

Suppose $\mathbb{Q} \cap [0,1]=\{r_1,r_2,r_3,...\}$ (We can do this because $\mathbb{Q}$ is countable and $\mathbb{Q} \cap [0,1] \subset \mathbb{Q}$. Let $x \in (0,1)$. Define ...
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On continuity of roots of a polynomial depending on a real parameter

Problem Suppose $f^{(t)}(z)=a_0^{(t)}+\dotsb+a_{n-1}^{(t)}z^{n-1}+z^n\in\mathbb C[z]$ for all $t\in\mathbb R$, where $a_0^{(t)},\dotsc,a_{n-1}^{(t)}\colon\mathbb R\to\mathbb C$ are continuous on ...
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calculus continuity of a hard question?

How do i calculate the continuity of a function if the functions are in a given set limit. I tried doing it but I epically failed… Please help!how do I solve it? When I tried to solve this I got that ...
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Krylov-Bogoliubov theorem without continuity

This question is very closely related to: Continuity in the Krylov-Bogoliubov theorem. The standard counterexample, which is presented in Katok-Hasselblatt is the following: Let $f:[0,1]\rightarrow ...
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Continuous injection from $\mathbb{R}^2$ to $\mathbb{R}^2$.

Could someone give me an example of a continuous injection from $\mathbb{R}^2$ to $\mathbb{R}^2$ which does not have a continuous inverse.
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35 views

continuity with lipschitz dertivative

Can somone explain me what does it mean to say a "function is continously differentiable with a Lipschitz derivative near the limit point". I dont understand the technical jargoans. Thanks
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limit of limit superior w.r.t truncated set

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
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Proving a fact about continuous function

Prove that if $f(a)>0$ and $f$ is continuous, then there is a $\delta >0$ such that for all $x$, $|x-a|< \delta$ implies $f(x)>0$.
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Is the function continuous at the indicated point?

$f(x) = x[x]$ at $x=2$ ($[x]$ is the greatest integer function) I am little confused, it seems like the function does exist at the given point. when limit goes to 2, $2[2] = 4$ and $f(2) = 4$ so ...
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If the function $f(x) =\lfloor \frac{(x-2)^3}{a}\rfloor \sin(x-2) +a\cos(x-2), \lfloor . \rfloor$ denotes …

Problem : If the function $$f(x) =\left\lfloor \frac{(x-2)^3}{a} \right\rfloor \sin(x-2) + a \cos(x-2),$$ where $\lfloor . \rfloor$ denotes the greatest integer function is continuous and ...
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1answer
24 views

How to show $F(\vec{x})$ is lipschitz on $[0,1] \times [0,1]$

We know that $f(x)=x^2$ is lipschitz on [0,1]; intuitively the steepest the graph of $f$ gets is at $x=1$, and I can find the lipschitz constant by a simple factoring argument. Now say ...
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694 views

A function takes every function value twice - proof it is not continuous

I want to prove the following nice statement I've found: A function $f: [0,1] \rightarrow \mathbb{R}$ takes every function value twice - proof it is not continuous I've already found an answer to my ...
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1answer
44 views

Showing a function is limited

This is something about which I'm quite confident, but I would really like to be sure Let $f(x)$ be a continuos function in $\mathbb{R}$ Then if $\displaystyle \lim_{x \to \pm\infty} f(x) = l \in ...