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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Show that $|sin(x)+cos(x)|$ is continuous at $\pi$

Show that the function $f(x)= |\sin(x)+\cos(x)|$ is continuous at $x=\pi$. By drawing the graph, we can easily show that it is continuous, but how can we show it by using limits. Please help.
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1answer
26 views

A continuous integer-valued function on a compact metric space has finite range

Let $X$ be a compact metric space and let $f:X\to\mathbb Z$ be a continuous function. (Here $\mathbb Z$ has the Euclidean topology induced from $\mathbb R$.) Prove that $f$ can assume only finitely ...
2
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0answers
43 views

Continuity in the complex plane

I was reading a book where it is claimed that a sufficient condition for \begin{equation} f(x)=\frac{1}{2\pi}\left|\sum_{j=0}^{\infty}\theta_je^{ix j}\right|^2 \end{equation} to be continuous and is ...
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4answers
57 views

Is a continuous bijection function from a hausdorff space to a compact space is a homeomorphism?

We know a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism. But I am wondering what happened if we switch the domain and codomain. Is a continuous bijection ...
2
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0answers
23 views

Function continuous at irrationals and discontinuous at rationals [duplicate]

Q: Given the function $f(x)=\sum_{n=1}^\infty f_n(x)$, where $f_n(x)= \left\{ \begin{array}{lr} 0; \;\;if \;x< r_n \\ \displaystyle \frac{1}{2^n}; x\geq r_n ...
0
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1answer
52 views

If a limit does not exist does that make it unequal to some given value?

I was asked to pick a function $f$ for which $\lim_{x\to c^-} f(x) \neq \lim_{x\to c^+} f(x)$ for some $c$. I used $f(x)=\sqrt{x-2}$ with $c=2$ as an example of such a function. My question is the ...
1
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1answer
37 views

Show that $f(a)=f(a+\frac{1}{n})$ for some $a \in [0, 1-\frac{1}{n}]$, given that:

Show that $f(a)=f(a+\frac{1}{n})$ for some $a \in [0, 1-\frac{1}{n}]$, given that: $f$ is continuous on $[0,1]$ and $f(0)=f(1)$. $f(a)=f(a+\frac{1}{2})$ for some $a \in [0, 1/2]$. $n\in \Bbb N$ and ...
0
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2answers
51 views

Does this piecewise function contradict the fact that all differentiable functions are continuous?

I learned that all differentiable functions are continuous. Why does the following equation not violate this rule: $$f(x)=\begin{cases}x^2+3 \quad &\text{when } x>1 \\ x^2 \quad ...
2
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1answer
47 views

How to use Cauchy-Scharwz inequality to prove differentiable?

I'm attempting to understand how to prove the function f such that $$f(x,y)=\frac{x^3y}{x^4+y^2}\;if\;(x,y)\neq (0,0)$$ $$f(x,y)=(0,0)\;if\;(x,y)=(0,0)$$ is continuous in $\mathbb R^2$. The solution ...
0
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1answer
56 views

About the continuity of $f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k}$

Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$. Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} ...
3
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0answers
32 views

Proposed proof of continuous operator on Sobolev space

Hi I am interested in a question about continuity: Assume that $\Omega \subset \mathbb{R}^{n}$ is bounded and consider operator $$f:W^{1,p}(\Omega) \times L^{p}(\Omega;\mathbb{R}^{n}) \rightarrow ...
4
votes
2answers
72 views

Prove that $f $ is constant

Let $f:\mathbb R \to \mathbb R $ be a continuous function such that for all $x \in \mathbb R$, $f(x)=f(x^2) $ prove that $f$ is constant. "please give me hints not answer. thanks a lot. :):):):):)" ...
0
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1answer
24 views

Finding continuous functions from a set

Let $A=\{0,1,\frac{1}{2},\frac{1}{3},...\}$. I want to find continuous functions from $f:A\to \mathbb R$. I proceed in this way. Any sequence converges to $x(\neq 0)$ will be eventually constant ...
3
votes
3answers
74 views

prove $f(x)=x$ has a unique solution

Question: Let $f$ be a continuous function from $\mathbb{R^2} \rightarrow \mathbb{R^2}$ such that $| f (x)− f (y)| ≤ \frac {1}{3} |x−y|$. Prove $f(x)=x$ has a unique solution. My sketch: There ...
2
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0answers
38 views

Shannon Entropy Continuity Constraint

I have the following problem: I want to find the probability density $p$ which maximizes the Shannon entropy \begin{equation} S := - \int_{x_b}^{x_c} dx ~ p(x) \log (p(x)) \end{equation} under the ...
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1answer
47 views

The set of all fixed points of a continuous function $f:[0,1] \to [0,1]$ , satisfying $f \circ f=f$ , is a non-empty interval ? [closed]

Let $f:[0,1] \to [0,1]$ be a continuous function such that $f \circ f=f$ on $[0,1]$ , then is it true that the set $\{x \in [0,1] : f(x)=x \}$ is a non-empty interval ?
1
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2answers
35 views

Prove that $f(x)$ is bounded. Please check my proof.

Assume $f:(0, \infty) \rightarrow \mathbb{R}$ is continuous. Also assume $\lim_{x \rightarrow 0}f(x)$ and $\lim_{x \rightarrow \infty} f(x)$ exist and are finite. Prove that $f(x)$ is bounded. ...
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3answers
90 views

Continuity is required for differentiability?

My professor emphasized that: Differentiability implies continuity and Continuity is required for differentiability. Since a function like $\frac 1 x$ is differentiable but not continuous, I ...
1
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1answer
62 views

Let $f(x)=(x+1)^2$. Prove that f is continuous at 0

I've started work from the definition, so for all $ϵ>0$, there is $δ>0$ such that $0<|x|<δ$, then $|(x+1)^2-(0+1)^2|<ϵ$. Then by expanding, $|x^2+2x|<ϵ$, $|x||x+2|<ϵ$, and by ...
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2answers
35 views

uniform convergence of sequence of function

I have a sequence of function $f_n$: $$ f_n(x) = \sqrt{x^2 + \frac1n} \qquad \text{on the interval } [-1,1] $$ and $$f(x) = |x| $$ I need to prove that the sequence of functions $f_n$ is uniformly ...
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0answers
15 views

Showing a map is a bounded linear operator.

Show that the map A : (C[0,1],∥·∥∞) → R, Ax = x(0), ∀x ∈ C[0,1] is a bounded linear operator. I know one has to show the map is continuous but I'm not sure how to go about proving it in this case. ...
0
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0answers
25 views

continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following: Does a continuous and concave function \begin{eqnarray*} f: N_{\mathbb{Q}} \to \mathbb{R} ...
1
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1answer
39 views

Prove that this integral is diverge

Let $f:[0,\infty) \to \mathbb R$ be a strictly decreasing continuous function, such that $\lim_{x \to \infty}f(x)=0$ prove that $\int_{0}^{\infty}\frac{f(x)-f(x+1)}{f(x)}$ is diverge.
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0answers
19 views

proof of continuous function for any real x

I have a function : $$ \sum_{n=1}^\infty \frac{sin(nx)}{n^2} \cdot x^2 $$ How is this function a continuous function for any $x \in \mathbb R $? I cannot prove it..
0
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1answer
23 views

uniform convergence of a function (continuous or differentiable or both?)

I have a function $S$: $$ S(x) = \sum_{n=1}^\infty \frac1{x+n^2} \\ \text{for} \ x \ge 0 $$ I need to determine if $S$ is continuous or differentiable or both.
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2answers
37 views

Proving continuity of a function at a point - Homework

$\Bbb R^2$ is using the Euclidean metric, $\Bbb R$ is using the standard $|y-x|$ metric. We define $f:\Bbb R^2\rightarrow\Bbb R$ by $$f(x,y) = \left\{\begin{array}{ll} \frac{x^6+y^6}{x^2+y^2} & ...
2
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1answer
24 views

Construct a sequence of continuous functions which converges pointwise to $\lfloor x \rfloor$

Suppose $f(x)=\lfloor x \rfloor$ for $x \geq 0$. Define a sequence of functions $(f_n(x))_{n \geq 1}$ where $f_n(x) = \left\{ \begin{array}{lr} x^n & : x \in [0,1)\\ (x-1)^n+1 ...
2
votes
2answers
41 views

Differentiability/continuity of piecewise defined functions

Let $$f(x)=\begin{cases}x^2\sin(\frac{1}{x}), &x\not= 0,\\ 0, &x = 0.\end{cases}$$ Since I can differentiate both parts of this, technically, $f$ is differentiable for all $x$. However I have ...
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36 views

Proving that a function is not continuous

Let $f :\mathbb R \to \mathbb R$ be defined by $f(x) = 1 − x$ when $x > 0$, or $0$ when $x ≤ 0$. Prove from the definition that $f$ is not continuous at $0$. Progress I wrote down a negation of ...
1
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0answers
24 views

Product of discontinuous and smooth function

Let $f$ be a smooth function (i.e. $C^\infty(\mathbb{R}^n)$), $g$ discontinuous in some $x_0\in \mathbb{R}^n$ and smooth everywhere else, for example $g(x)=\Theta(x)$ or $g(x)=\frac{1}{x}$. ...
0
votes
1answer
25 views

Show Open/Closed for a Set and two continuous functions

Let f, g : X → R be two continuous functions defined on a metric space X. (i) Show that the set U = {x ∈ X : f(x) > g(x)} is open in X. (ii) Show that the set F = {x ∈ X : f(x) ≥ g(x)} is closed in X. ...
0
votes
1answer
30 views

Two-variable limit problem: the limit of $(\cos^2(\sqrt{x^2+y^2})-1)/(x^2+y^2)$ as $(x,y)\to 0$

What is the value of $k$ such that $f$ is continuos in $(0,0)$? $$f(x,y) = \begin{cases} \dfrac {\cos^2\left(\sqrt{x^2+y^2}\right)-1}{x^2+y^2}, & \text{if $(x,y)$ $\ne$ (0,0)} \\[2ex] k, & ...
1
vote
3answers
36 views

continuous function

$$g(x) = \left\{\begin{array}{cl} x\sin\left(\frac{\cos(x)}{x}\right) & \text{if } x \neq 0\\ 0 & \text{if } x=0\end{array}\right.$$ Show that this function is continuous at $x=0$. so the ...
1
vote
1answer
28 views

Show that $f$, defined on a closed cover of $X$, is continuous.

Let $\mathbb{B}$ be a finite closed cover of a topological space $X$. For each $B \in \mathbb{B}$, let $f_B: B \to Y$ be continuous. Furthermore, suppose for each pair $A, B \in \mathbb{B}$, $f_A|_{A ...
0
votes
1answer
25 views

Convergence of integral of multiplication of two positive functions

I have two functions $f, g:\mathbb{R}\rightarrow \mathbb{R}_{\ge0}$, that are continuous. I know that $\int\limits_{-\infty}^\infty f(s) \, ds=C_1<\infty$, and $g(s)\le C_2$, with $C_1> 0$ and ...
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2answers
52 views

Why does differentiability implies continuity, but continuity does not implies differentiability?

Why does differentiability implies continuity, but continuity does not implies differentiability? I am more interested in the part about a continuous function not being differentiable. Well, all ...
0
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0answers
23 views

When does a continuous function defined on a closed and bounded convex set has a fixed point?

For a function $f$ defined from a domain $K$ to itself, we have a point $x$ in $K$ is said to be a fixed point of $f$ if $f$ maps $x$ to itself. When the domain K is a compact convex set with some ...
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2answers
31 views

Limits in matrix norm if convergence of integration is guaranteed

the answer of this question probably is very obvious, but I want to make sure this is correct. I have a function $F: \mathbb{R}\rightarrow \mathbb{R}^{n\times n}$ that is continuous and ...
3
votes
1answer
49 views

If $U(f,P) = L(f,P)$, show that $f$ is constant.

The question has two part, Show that if f : [a, b] → R is continuous and there exists a partition P of [a,b] such that U(f,P) = L(f,P), then f is constant. Is this true if we drop the assumption ...
0
votes
4answers
46 views

Finding Continuous Functions

Find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x \in \mathbb{R}$, $f(x) + f(2x) = 0$ I'm thinking; Let $f(x)=-f(2x)$ Use a substitution $x=y/2$ for $y \in ...
1
vote
3answers
27 views

Chain of Implications for Continuity and Boundedness

Consider the following definitions: > 1). Somewhere Locally Bounded: $\exists p \in X, \exists \epsilon >0, \exists \delta >0, \forall q \in X: d(p,q)< \delta \Rightarrow ...
1
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2answers
40 views

[edited]Prove that $f(x)=0$ exists in a certain interval.

I have $f:R \rightarrow R$, $f(0)=-1$ and $f'(x) \ge1$ $\forall x$. I need to show that $f(x)=0$, for some $x\in[0,1]$ I know that I need to use mean value theorem and intermediate value theorem. ...
0
votes
1answer
38 views

Continuity of unique solution to differential equation

Let $f$ be a continuous function on $G$, where $G \subseteq \mathbb{R}^2$ is an open set containing $I \times [a,b]$ where $I:=[x_0-d,x_0+d]$, for some $a,b,d \in \mathbb{R}$ s.t. $a<b$ and ...
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votes
1answer
29 views

Examples of unbounded continuous function $f:Q\cap[0,1]\rightarrow R$

I want to find examples of unbounded continuous function $f:Q\cap[0,1]\rightarrow R$ I am thinking $\frac{1}{1+n}$ may satisfy but not quite sure. And if there are I want to see other examples too. ...
0
votes
1answer
30 views

Help with proof that ${1\over x+1}$ is continuous at point a=0

Hello I'm struggling to prove that the function ${1\over x+1}$ is continuous at the point a=0. (The function has -1 excluded from its domain). I understand for any $\varepsilon$ we must pick a ...
2
votes
1answer
40 views

Show that $ f(x)=x^4 $ is continuous at the point $x=-7$

Show that $ f(x)=x^4 $ is continuous at $x=-7$. Proof: Using the $\epsilon$-$\delta$ definition I get the following: $$|x+7|<\delta \implies |x^4-2401|<\epsilon $$ ...
0
votes
0answers
24 views

Second derivative relating to continuity

Given a function $f$ such that the integral $A(x)=\int_a^x{f(t)dt}$ exists in an interval $[a,b]$. Let $c$ be a point in the open interval $(a,b)$. Consider the following ten statements about this $f$ ...
1
vote
2answers
74 views

Bounded functions on a compact interval

If i have given $f:[0,1] \rightarrow \mathbb{R}$ $f$ is bounded. $g:[0,1] \rightarrow \mathbb{R}, x \rightarrow xf(x)$ And i have to prove $g$ continous in x=0. What can i say about $f$, is it ...
0
votes
1answer
24 views

piece wise functions continuity

Find the value of ‘$k$’ that makes the function $h(x)$ everywhere continuous $$h(x)=\begin{cases}x^2-k & x <-1\\ x^3+3x^2+1 & x\geq -1\end{cases}$$
19
votes
2answers
1k views

Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...