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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Continuity of a function $f: \mathbb{R}^2 \to \mathbb{R}$

It's easy to check that the function $$ f_1(x, y) = \begin{cases}\frac{x y}{x^2 + y^2} &\text{if (x, y) ≠ (0, 0)}\\0&\text{if (x, y) = (0, 0)}\end{cases}$$ is not continuous in $0$, because ...
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1answer
26 views

Implicit function theorem application: $h(f(v),v)=0$ find $f(v)$…

I am preparing for an exam and I cannot seem to figure out how to solve exercises where I need to apply the implicit function theorem. Exercise: Let $h: \mathbb{R}^2 \rightarrow \mathbb{R}$ given ...
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1answer
39 views

Question about Rudin's Functional Analysis Closed Graph Theorem

In page 51 of Rudin's Functional Analysis, the closed graph theorem is proven, which says that if you have a linear map between two F-spaces whose graph is closed in the product space, then the map is ...
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35 views

Completion of C(I) to $L^{2}(I)$ for some arbitrary interval I

As $L^{2}(I)$ is the completion of C(I), without too many issues (as $L^{2}(I)$ is a space of equivalence classes with equivalence relation defined as functions equivalent if differ at only finitely ...
2
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1answer
70 views

Is there a name of such functions?

Let $U$ be an open subset of $ \mathbb R^n$ and consider $f :\mathbb R^n \to \mathbb R$ with the properties that $ f( \partial U)=0$ and $f$ takes negative values on $U$. My questions: Is there ...
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1answer
43 views

What implies $f_n (y) \leq f(x) + \epsilon$ about $f$ ?

Let $X$ be a regular topological space. Question: For which functions $f : X \rightarrow \mathbb R$, can we find a sequence of functions $f_n : X \rightarrow \mathbb R$ such that: $\forall ...
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2answers
325 views

What is the definition of differentiability?

Some places define it as: If the Left hand derivative and the Right hand derivative at a point are equal then the funtion is said to be differentiable at that point. Others define it based on ...
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44 views

Bounded holomorphic function uniformly continuous.

Let $f:\{z\in\mathbb C\vert~ Re(z)>0\}\to\mathbb C$ be holomorphic and bounded. I would like to show that $f$ is uniformly continuous on $\{z\in\mathbb C\vert~ Re(z) > c\}$ for each $c>0$ but ...
4
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2answers
44 views

Inverse image of every compact set is compact under a function

Let $f:\mathbb R\to \mathbb R$ be a function such that $f^{-1}(K)$ is compact for every compact set $K$ in $\mathbb R$. Then $f$ is continuous $\sup \limits_ {\mathbb R}f(x)<\infty$ $\inf ...
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2answers
24 views

Does $\cup_{n=1}^{\infty} {h_n}^{-1} (\overline{U_n}) \subset \overline{ \cup_{n=1}^{\infty} {h_n}^{-1} (U_n) }$ hold?

Let $\{ h_n :X \to Y\}_{n \in \mathbb{Z^+}} $ be a sequence of continuous functions from a topological space $X$ to another topological space $Y$, and for each $n$ let $U_n$ be an open subset of $Y$. ...
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1answer
89 views

Finding Function's Extension and Its Unique Existence.

Let $$A= \left\{\frac j{2^n}\in [0,1] \mid n = 1,2,3,\ldots,\;j=0,1,2,\ldots,2^n\right\} $$ and let $$ f:A\rightarrow R $$ satisfy the following condition: There is a sequence $ \epsilon_n \gt 0 $ ...
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1answer
47 views

Show that the map $\epsilon \to z_{\epsilon}$ is continuous

Suppose that $f$ and $g$ are holomorphic in a domain containing the unit disc $D=\{z| |z| \le 1 \}$. Suppose that $f$ has a simple zero at $z=0$ and vanishes nowhere in the unit disc. Let ...
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1answer
50 views

Examine the continuity of function $f(x)=\frac{2x^2-4x}{|x+1|+|x-3|-2}$

Using the definition of absolute value for $$|x+1|=\begin{cases} x+1, & x\ge -1\\ -x-1, & x>-1 \end{cases}$$ and $$|x-3|=\begin{cases} x-3, & x\ge 3\\ -x+3, & x>3 \end{cases}$$ ...
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2answers
160 views

An example of a bounded, continuous function on $(0,1)$ that is not uniformly continuous

I can not find the example of a continuous function on $(0,1)$ that is bounded on $(0,1)$, but not uniformly continuous on $(0,1)$. Is there any? Thank you.
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2answers
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Let $ f:I \to \mathbb{R} , I=(0,1) $ be uniformly continuous. Then exists $ \lim_{n\to\infty} f(\frac{1}{n}) $

True. Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume: $$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$ Since $ \lim_{n\to\infty} \frac{1}{n} $ is ...
2
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1answer
253 views

Differentiability implies continuity - A question about the proof

I have a question, to aid my understanding, about the proof that differentiabiility implies continutity. Differentiability Definition When we say a function is differentiable at $x_0$, we mean that ...
0
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1answer
33 views

Uniform convergence implies continuity and differentiability?

For example: Suppose I have the following series: $$\sum_{k=0}^{\infty}e^{-k}\sin(kt)$$ The Weierstrass-M-Test shows that the series is uniformly convergent on $\mathbb R$. Does this imply ...
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2answers
43 views

Continuity and Uniform Continuity on half closed intervals

I have been stuck on the following problem for a long time : Prove that if a function $f:(a,b]\to\mathbb R$ is continuous, then it is uniformly continuous if and only if $\lim_{x\to a^+}f(x)$ ...
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0answers
48 views

Uniformly continuous functions on the interval [duplicate]

Let $f:[1,\infty)\to\mathbb R$ be uniformly continuous. Prove $\exists$ $M > 0$ s.t $$\frac{\big|f(x)\big|}{x} \leq M, \hspace{11pt} \forall x\in[1,\infty)$$
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1answer
43 views

Examine the continuity of a function $f(x)=\lim\limits_{n\to\infty}(x \arctan(n \cot(x)))$

If we know domain of function $\arctan(x),D_{1}=\mathbb{R}$ and $\cot(x),D_{2}=\mathbb{R}$ without $\{k\pi\}$ we need to check two cases: $x<0$ and $x>0$ How to evaluate limits in these cases? ...
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34 views

About functions and little calculus

Many a times I come upon an $x$ vs. $t$ graph in which the distance $x$ is given as a function of time like $x=f(t)=20+5t^2$. Can its reverse be found? For example, given ...
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2answers
40 views

The set of all limits of the image of a divergent sequence under a continuous function

Let $f:\mathbb R\to \mathbb R$ be a continuous function and let $A=\{y=\lim\limits_{n\to \infty}f(x_n):$ for some sequence $x_n\to \infty\}$. My intuition says that $A$ must be a singleton. But I have ...
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1answer
73 views

About the Heine-Cantor theorem.

I don't understand the Heine-Cantor theorem because of one example: The function $x\to \frac{1}{x}$ is not uniform continuous, and we can clearly see in the graph just by looking at the interval ...
2
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1answer
35 views

The functional is continuous

Show that the functional $J(y)=\int_a^b (\sin^3 x+y^2) dx$ is continuous in respect to the $||\cdot||_{\infty}$ norm, at any $y_0 \in C([a,b])$. Let $y_0 \in C([a,b])$. Then for $y \in C([a,b])$ we ...
2
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1answer
50 views

Closed graph theorem: How do domain and codomain affect continuity?

I had to examine the closed graph theorem under the following circumstances: $X, Y$ metric spaces with $Y$ compact. Does the theorem also hold if Y is not compact? (Assuming compactness in the ...
0
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2answers
36 views

Continuity and Differentiability of f(x)

$$f(x) = \begin{cases} x^2 + 3x + 2 & \quad \text{if } x \leq 0\\ x^2 - 3x + 2 & \quad \text{if } x > 0\\ \end{cases} $$ Prove that f is continuous at $x = 0$ and not ...
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2answers
36 views

Show that for any numbers $p$ and $q$, $\{f\in C[a,b]:p\leq f(x)\leq q\}$ where $x\in [a,b]$ is a closed subset of $C[a,b]$. Similarly for $L_2[a,b]$.

Show that for any numbers $p$ and $q$, $\{f \in C[a,b] \mid \forall x\in [a,b]: p\leq f(x)\leq q\}$ is a closed subset of $C[a,b]$. Similarly for $L_2[a,b]$. We must show that if $f_n\to F$ and ...
0
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1answer
32 views

f continuous and differentiable?

Consider the function $$f:\mathbb{R}^2\to\mathbb{R}\; (x,y)\mapsto \begin{cases} \frac{x^ay^b}{(x^2+y^2)^c}, & (x,y)\not=(0,0)\text{,}\\ 0, & \text{else } \end{cases}$$ I am trying to ...
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1answer
52 views

Relation between roots of a function and roots of its derivative, IVP

I am troubled with this question of my book: I do know that f (a) = f '(a) = 0 if the multiplicity of root 'a' is greater than 2 but how that fact is exploited here or is there something more ...
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1answer
79 views

How do I analyze/determine the continuity of a function?

My question is really:the following: In general, how do I analyze/determine the continuity of a function? Is there some sort of algorithm? Failing that, here's an example. $$ f: \left]-1,1\right[ ...
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The functional is not continuous in respect to the strong norm

Let $V=C^1([a,b])$. If $J$ is a continuous functional for the norm $\|\cdot\|_\infty$ then it is continuous for the norm $\|\cdot\|_1:= ||y||_{\infty}+||y'||_{\infty}, y \in V$. But the converse is ...
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1answer
20 views

Distance attained by a function

Let $A$ be a subset of $\mathbb R^n$ and let $x\in \mathbb R^n$. Then $\exists y_0\in A$ such that $d(x,y_0)=d(x,A)$ if $A$ is a non-empty subset of $\mathbb R^n$. $A$ is a non-empty closed subset ...
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2answers
46 views

In order to show that a function is C^1 is it enough to show that the 1. partial derivatives exists?

Hello I am having some issues with the following exercise: Let $\textbf{h}: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $\large \textbf{h}(u,v) = u^2 + (v-1)^2 - 5 + e^{u-2}$ (i) Show that ...
3
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1answer
21 views

A Continuous Choice of $k$-Subspaces of a Vector Space Gives a Continuous Choice of Bases

$\newcommand{\R}{\mathbf R}$ The Grassmannian $G_k(\R^n)$ as a topoplogical space is defined in the following way: Let $F_k(\R^n)$ be the collection of all the linearly independent lists of size $k$ ...
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2answers
77 views

If $f$ is continuous on $[a,b]$ then $1/f$ is bounded on $[a,b].$

$f(x) > 0$ is given for all $x\in [a,b]$. I only got to this: Let $c$ belong to $[a,b]$. Then, for all $ε>0$, there exists $δ>0$, such that, $|x-c|<δ\implies|f(x)-f(c)|<ε$.
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1answer
56 views

Continuity of the maximum of finite continuous functions

Let $(X,\tau)$ be a topological space and let $f_1,\ldots,f_n:X\to\mathbb{R}$ be continuous functions (the topology of $\mathbb{R}$ is the usual one). Define $g:X\to\mathbb{R}$ by ...
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1answer
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Two problems related to continuity of a metric from Munkres' topology book

Let $X$ be a metric space with metric $d$. Show that $d:X\times X\to \mathbb{R}$ is continuous. Let $X^\prime$ denote a space with the same underlying set as $X$. Show that if $d:X^\prime\times ...
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Prove that $f_n(x)$ is discontinuous at $x = 0$.

I am having problems with the following exercise, I am not sure if my procedure is correct. Exercise: Let $ \large f_n(x)=\left\{ \begin{array}{ll} 0 ~~~if~~x = 0 ...
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1answer
47 views

Are continuous functions dense in $L^1$?

It is a well known fact that the continuous compactly supported functions are dense in $L^1(\mathbb R)$. An immediate counterexample to this fact for a non locally compact space is $\mathbb R ...
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1answer
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Absolute continuity and sample paths of Brownian motion

An offhand remark in Morters and Peres' book on Brownian motion says that Brownian motion is a.s. absolutely continuous on compact intervals (see page 147, immediately preceding the statement of ...
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1answer
100 views

Continuously extending a set of independent vectors to a basis.

Question: Let $I=(a,b)$ be an interval and let $$v_i:I\to\mathbb{R}^n,\quad i=1,\ldots,k$$ be continuous curves such that $v_1(t),\ldots,v_k(t)$ are linearly independent in $\mathbb{R}^n$ for ...
3
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1answer
62 views

$f\circ g$ continuous, $f$ local homeomorphism, $g$ continuous in a different topology $\implies g$ is continuous

I've asked this question before but neglected some assumptions and got a less than useful answer as a result, so I'm going to try again. Let $g:I\times I\to Y$ (where $I=[0,1]$) be a function such ...
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1answer
60 views

Problem of Real analysis, continuous functions.

Problem: Let $f$: $\mathbb{R} \to \mathbb{R}$, growing funtion and $D(f)=\{t \in \mathbb{R} : f $ is not continuous in $t \}$. Show that: a) Exist $q: D(f) \to \mathbb{Q}$ such that for all $t \in ...
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2answers
113 views

If $f(x)$ is continuous at $a$ and $g(x)$ is not continuous at $a$, then can $(f+g)(x)$ be continuous at $a$?

I know that if both $f(x)$ and $g(x)$ are continuous at $a$, then $(f+g)(x)$ would be continuous at $a$. My first thought here is that $(f+g)(x)$ cannot be continuous at $a$ if $g(x)$ is not ...
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3answers
123 views

Prove that the trigonometric function is uniformly continuous

In my assignment I have to prove that the following function is uniformly continuous in $(0,\frac{\pi}{2})$: $$f(x)=\frac {1-\sin x}{\cos x}$$ Here is my suggestion for solution. Please let ...
3
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1answer
58 views

Proving a statement about a continuous function for which $\forall x\in\mathbb{R},\exists y>x : f(y)>f(x)$

Suppose $f$ is a function which is continuous on $\mathbb{R}$. Also, for all $x\in \mathbb{R}$, there exists $y>x$ such that $f(y)>f(x)$. I must prove that if $\lim_{x\to\infty} f(x)=L$ then ...
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1answer
44 views

Integral of $au^2$ where $a$ is continuous and $u \in W_0^{1,2}(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a smooth domain with bounded complement. Let $a \in C(\Omega)$ and let $u \in W_0^{1,2}(\Omega)$. Suppose that $a > 0$ in $\Omega$ and $\displaystyle ...
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6answers
64 views

Fundamental Theorem of Calculus application for $f(x)\geq 0$

Can anybody help me with how to solve the following question using the fundamental theorem of calculus? I'm a bit confused... If $f$ is a continuous function on $[a, b]$ and $f(x)\geq 0$ for all ...
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1answer
42 views

Continuity in closed sets

Please help me, I have being trying this for days now. Let $f:F \to \mathbb{R}$ be a function on a closed set $F$. Show that $f$ is continuous if and only if $A=\{x \in F; f (x) \leq c\}$ and $B=\{x ...
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1answer
70 views

Non-continuous topology?

I've been studying topology this term and it really got me interested. But sometimes in math I feel that we are just taught things one by one, without really talking about why we do it that way. So I ...