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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Continuous function on a compact set with no fixed points

I'm reviewing this problem for my analysis qual. Let $f:X\rightarrow X$ be a continuous mapping from a metric space to itself. Assume $f $ has no fixed points. Prove that, if $X $ is compact, ...
6
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2answers
239 views

Is a function defined at a single point continuous?

Is a function defined at a single point continuous? For example $f:\{0\}\to\{0\}$ defined by $f(x)=\sqrt{x}+\sqrt{-x}$ is a sum of two continuous functions and is therefore continuous, however for ...
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2answers
58 views

To check continuity of Multivariable functions

To check continuity of function at origin given by $$F (x, y) = \begin{cases}\dfrac{xy^{2}}{x^{2} + y ^{4}}&;& \mbox{otherwise},\\ 0&;&\mbox{ at origin}. \end{cases}$$
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1answer
54 views

Approximating continuous functions on a product space

I have read in a paper that all bounded continuous functions $$f:[0,t] \times S \to \Bbb R$$ can be uniformly approximated by functions of the form $$f(s,w) = \sum_{m=1}^{N}a_mg_m(s)f_m(w)$$ (for ...
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1answer
44 views

Continuty of functions

We have the following situation. Assume that $X$ a topological space, $u\in C(X)$ and $u\geq0$, then $U:=\{x\in X: u(x)>0\}$ is open in $X$. Moreover let $f\in C(U)$ and define for each ...
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4answers
92 views

Sufficient condition for convexity

Let f:$ [a,b] \rightarrow \mathbb{R} $ a continous function such that $ \forall (x,y) \in [a,b]^{2}, \exists t \in ]0,1[, f(tx+(1-t)y) \le tf(x) + (1-t)f(y) $ show that f is convex
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2answers
49 views

A question about two common definitions

Two definitions make me puzzled ! 1. The definition of $\textbf{Functions Differentiable at a Point}$: A function $f$ defined in a neighborhood $(x_{0}-\delta,x_{0}+\delta)$of a point $x_{0}$, ...
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1answer
154 views

Definition of continuity in topological spaces does not seem quite right.

Here's how continuity is defined in most standard topology texts A function from $X$ to $Y$ is continuous iff the inverse image of each open set of $Y$ is open in $X$. This definition does not ...
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1answer
153 views

For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $

Suppose that $f$ is continous on $[0,1]$ , differentiable on $(0,1)$ , and $f(0)=0$ and $f(1)=1$.For every integer $n$ show that there must exist $n$ distinct points ...
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0answers
63 views

Intersection of two continuous functions

Suppose that $f:[0,1]\to\Bbb{R}$ is continuous and that $f(0)=f(1)$. Show that there is some $p\in [0,1]$ such that $f(p)=f(p+1/2)$ I drew some graphs and tried to show that ...
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0answers
15 views

Minimum condition needs to add to make a continuous extension

$f$ is a continuous function from $S$ to $\Bbb R$. What is the minimum condition needs to be added to make the function $$f : \bar{S} \to \Bbb R$$ have a continuous extension. Thanks in advance
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1answer
61 views

Continuity in $\mathbb R^n$

We know that continuity along all directions does not imply that the function is continuous in multivariate space. Intuitively is it right to think that a function can be discontinuous along a ...
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0answers
14 views

Upper hemicontinuity of a correspondence

I would like to know whether the following correspondence is upper hemicontinuous: $$ C(x)=\begin{cases} 1, & (f(x)>0) \\ [0,1], & (f(x)=0) \\ 0, & (f(x) < 0) \end{cases}, $$ ...
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1answer
59 views

Continuous functions and existence of a root

Let $\, f:[1,2] \rightarrow \mathbb R$ be a continuous function such that for every $n$ $\in$ $\mathbb N, \exists$ $x \in [1,2]$ with $\ |f(x)| < \frac 1n$ Show that $ \exists \;c \in [1,2]$ such ...
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3answers
198 views

Uniform continuity of a function $F:(0,1)\times (0,1)\to \mathbb R$

Let, $f,g:(0,1)\times (0,1)\to \mathbb R$ be two continuous functions defined by $f(x,y)=\dfrac{1}{1+x(1-y)}$ and $g(x,y)=\dfrac{1}{1+x(y-1)}$. Then which is correct? $f$ and $g$ both are uniformly ...
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1answer
21 views

Uniform continuity 2 variable function

Let $f:\Bbb{R^2}\rightarrow\Bbb{R}$ and two real parameters $a,b$ such that $$f (\mathbf{x,y})= \begin{cases}a(x^2+y^2),&(x,y)\in B_{d_{2}}(0;2)\\ \frac{b}{\sqrt{x^2+y^2}},&(x,y)\in \Bbb{R^2}- ...
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0answers
43 views

Continuity of two variable function with compactness and supremum norm

Please assist me with the following homework problem: Let $X$ and $Y$ be metric spaces and suppose that $Y$ is compact. Let moreover $f: X \times Y \to R$ be a continuous function, and define ...
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0answers
41 views

Cauchy Continuous function is continuous?? [duplicate]

Suppose $X$ and $Y$ be metric spaces.Let $f:X\to Y$ be function which is Cauchy Continuous.Show that $f$ is continuous. Cauchy Continuity:Let $X$ and $Y$ be metric spaces, and let $f$ be a function ...
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3answers
159 views

If a function is discontinuous on $\mathbb Q$, is it necessarily discontinuous on $\mathbb R \setminus \mathbb Q$?

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}$ is discontinuous on $\mathbb Q$. Is $f$ necessarily discontinuous on $\mathbb R \setminus \mathbb Q$?
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2answers
36 views

Show that the function f(x,y) maps the set $x^2+y^2 \leq 1$ in a closed set in $\mathbb{R}$.

Show that the function $$f(x,y)=\begin{cases} a+b{\tan(x^2+y^2)\over x^2+y^2} & (x,y) \neq (0,0) \\a+b & (x,y) = (0,0) \end{cases}$$ maps the set $x^2+y^2 \leq 1$ in a closed set in ...
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3answers
104 views

Why is a norm a continuous function? (Question about existing proof)

I'm trying to follow the proof given in this answer: http://math.stackexchange.com/a/265595/188401 I understand the proof in general, but I have a question. It's mentioned that "In this case it ...
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2answers
49 views

If $f$ is a continuous function from $R^3$ to $R$ and $K⊂R^3$ is compact, show that there exist two points $a, b ∈ K$ so that $f(K)⊂[f(a),f(b)]$

If $f$ is a continuous function from $R^3$ to $R$ and $K⊂R^3$ is compact, show that there exist two points $a, b ∈ K$ so that $f(K)⊂[f(a),f(b)]$. When is $f(K)=[f(a),f(b)]$? What I believe is the ...
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1answer
41 views

Density of $C_c(X)$ in $L^p$

1) If $X$ is a locally compact Hausdofrf space, then $C_c(X)$ is dense in $L^p(\mu)$; $\mu$ it is the measure obtained by the Riesz representation theorem. 2) Suppose that $X$ not is locally ...
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2answers
123 views

Continuous function on complete bounded metric space need not be bounded

I came across the following old qual problem: Suppose $(X,d)$ is a complete metric space with finite diameter. Is every continuous function on $X$ bounded? It seems like the function $1/x$ on ...
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1answer
32 views

Find the map of the closed ball $B(0,1)$ of the following continuous function $f(x,y,z)=(\frac x3,\frac y2-1,\frac z9+1)$ and $f^{-1}(0)$.

Find the map of the closed ball $B(0,1)$ of the following continuous function $$f(x,y,z)=\left(\frac x3,\frac y2-1,\frac z9+1\right)$$ and $f^{-1}(0)$. $f^{-1}$ seems quite simple, I got ...
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1answer
26 views

Proving linearity implies (or can imply under opportune conditions) lower semicontinuity

A function $f:X\to\mathbb{R}$, with $X$ being a topological space, is termed as lower semicontinuous (lsc) at $x_0\in X$ if: $$\forall\epsilon>0\,\,\exists V\text{ an open neighborhood of }x_0:x\in ...
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2answers
189 views

When is a continuous function also a bijective function?

To begin with, I would like to set forth a property of continuous functions: There doesn't exist a continuous function $f$ on $\mathbb{R}$ such that $f|_{\mathbb{R}\setminus \mathbb{Q}} : ...
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1answer
47 views

The following is unclear to me:($f$ is the identical function) the inverse of $f: X\to X$ is not continuous(if the second space isnt discrete)

My book defines a function $f:X\to Y$ as continuous in $a\in X$ if for every neighbourhood $V$ of $f(a)\in Y$ exists a neighbourhood of $a$ so that $f(U) \subset V$. The equivalent : if $V$ is the ...
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1answer
45 views

Pointwise and Uniform convergence with one-sided limit

Consider a function $g(x,y)$, $x\in X$ and $y\geq 0$ where $X$ is a compact subset of $\mathbb{R}$. Assume that $g(x,y)$ converges pointwise to zero as $y\downarrow 0$, for all $x\in X$. Is the ...
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1answer
68 views

Find all continuous functions $f:\mathbb R\to\mathbb R$ such that $f(f(x))=e^{x}$ [duplicate]

Find all continuous functions $f:\mathbb R\to\mathbb R$ such that $f(f(x))=e^{x}$ I didn't solve this problem, but I proved that f is increasing and $ x<f(x)<e^{x} $ please help
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3answers
294 views

Can a function be continuous but not Hölder on a compact set?

Is it possible to construct a function $f: K \to \mathbb{R}$, where $K \subset \mathbb{R}$ is compact, such that $f$ is continuous but not Hölder continuous of any order? It seems like there should ...
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1answer
45 views

Showing that $\inf\left|x-p\right|$ is continuous

Let $X\subset R$ be non-empty. $f :\Bbb{R}\to\Bbb{R}$ is defined by $$f(p)=\inf_{x\in X}\left|x-p\right|$$ for every $p\in\Bbb{R}$. How do I show that $f$ is continuous? I tried using the reverse ...
4
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1answer
52 views

Showing that $g$ is continuous by showing that a series converges?

Define $g:\Bbb{R}\setminus\Bbb{Z}\to\Bbb{R}$ by $$g(x)=\dfrac{1}{x}+\sum\limits_{n=1}^\infty\dfrac{2x}{x^2-n^2}.$$Show that $g$ is continuous. $g$ can be rewritten as ...
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1answer
29 views

Continuous bijection at boundary of open set

Suppose $f:U \to V$ is a continuous bijection, where $U \subset \mathbb{R}^n$,$V \subset\mathbb{R}^m$ and $U$ is open. Suppose further that $U \ni x_n \to x \notin U$. Then $y_n:=f(x_n)$ may not ...
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2answers
58 views

$f$ continuous identical to absolutely continuous $g$ a.e. $\Rightarrow f=g$?

Let $f:[a,b]\to\mathbb{C}$ be a function identical to an absolutely continuous function $g$ almost everywhere. I was wondering whether if $f$ is continuous we can infer that $f$ is absolutely ...
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1answer
55 views

Continuity and Convergent nets. [duplicate]

The original question before it was marked as duplicated question: I was wondering if one construct a non-continuous function $f:X\to Y$ between two topological spaces $X$ and $Y$, such that $f$ ...
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What is the topological interpretation of continuity of distributions?

I was given this definition of continuity in the distributional sense. A distribution $T$ over the space of test functions $\mathcal{D}$ is continuous if for every sequence of test functions $\{ ...
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2answers
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Find a function such that $f\notin C^1(\mathbb{T})$ but it's Fourier series converges to it uniformly

Find a function such that $f\notin C^1(\mathbb{T})$ but it's Fourier series converges to it uniformly So (I think) if $f$ is $C^1(\mathbb{T})$ then there's a theorem says that it's Fourier ...
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1answer
45 views

Prove that $f\ast g$ is continuous if $f\in C(\mathbb{T})$ and $g\in R(\mathbb{T})$

Prove that $f\ast g$ is continuous if $f\in C(\mathbb{T})$ and $g\in R(\mathbb{T})$ (Meaning $f$ is continuous and periodic and $g$ is Riemann integrable and periodic). So basically, if we define ...
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1answer
32 views

strongly continuous mapping implies bounded mapping

Hi does anyone know how to show the result that if we have a relexive Banach space $X$ and a mapping $A: X \rightarrow X^{*}$ (not necessarily linear), which is strongly continuous, which means ...
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0answers
54 views

For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
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1answer
85 views

Is continuous $f$ constant if every point of $\mathbb{R}$ is local minimum of $f$?

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous. Is $f$ constant if every point of $\mathbb{R}$ is local minimum of $f$? What metric spaces we can use instead of $\mathbb{R}$? I guess we ...
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1answer
41 views

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?
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57 views

If every function $f:X\to\mathbb R$ is continuous for a non-empty topological space $X$ then does $X$ have the discrete topology ?

Let $X$ be a non-empty topological space such that every function $f:X\to\mathbb R$ is continuous , then is every subset of $X$ open ?
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1answer
62 views

Inequality for line integral

Let $F(x)$ be a continuous (not necessarily monotonic) function defined on smooth curve $C$. I am wondering if the following inequality holds for line integrals. $$|F(a)-F(b)|\leq \int_C ...
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0answers
23 views

Proof of converse of Intermediate Value Property [duplicate]

I need to prove that if a function is injective(one-one) and has Intermediate Value Property, then the function is continuous. I just need a hint as I would like to solve it on my own. P.S.-Remember ...
0
votes
2answers
92 views

Proving that if the sum of monotonic increasing functions is continuous in a point then each one of them is continuous in the same point.

$g(x)$ and $h(x)$ are monotonic increasing functions s.t the function $(g+h)(x)$ is continuous in $x_0$. prove that $g(x)$ and $h(x)$ are continuous in $x_0.$ here are some thoughts that hit me: I ...
0
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2answers
53 views

Proving that if $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous in $“0”$ and fulfills $f(x)=f(2x)$ for each $x$ $ \in\Bbb{R}$ then $f$ is constant.

How do I prove that if $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous in $"0"$ and fulfills $f(x)=f(2x)$ for each $x$ $ \in\Bbb{R}$ then $f$ is constant.
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0answers
90 views

An issue with Weierstrass theorem's proof (extreme value theorem)

I'm having some issues while I try to understand Weierstrass theorem's proof. Theorem If a real-valued function $f$ is continuous in the closed and bounded interval $[a,b]$, then $f$ must attain ...
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1answer
32 views

Given $|g(x)|\leq M|x-2|$, prove that g is continuous at $x=2$

Given $|g(x)|\leq M|x-2|$, prove that g is continuous at $x=2$ I am not sure whether the way I solve it is correct or not. Given any $\epsilon>0$, take $\delta=\frac{\epsilon-|g(2)|}{M}$ ...