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

$$g(x) = \left\{\begin{array}{cl} x\sin\left(\frac{\cos(x)}{x}\right) & \text{if } x \neq 0\\ 0 & \text{if } x=0\end{array}\right.$$ Show that this function is continuous at $x=0$. so the ...
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1answer
33 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 ...
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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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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 ...
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36 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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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 ...
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1answer
77 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 ...
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4answers
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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 ...
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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 ...
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2answers
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[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. ...
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1answer
60 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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1answer
35 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. ...
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1answer
33 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 ...
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1answer
46 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: $$|x+7|<\delta \implies |x^4-2401|<\epsilon $$ ...
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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 ...
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piece wise functions continuity

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

Example of continuous functions $f\colon S \to T$ such that $f(S)=T$.

I would like to find an example of a continuous function from $S=(0,1)$ to $T=(0,1)\cup (1,2)$ such that $f(S)=(T)$. At the moment the only thing I can think might work would be to check whether ...
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3answers
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If $f: \mathbb{R} \to \mathbb{R}$ is continuous then $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$

Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$. At first I thought this was quite ...
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1answer
24 views

some statements based on continuity

$f:\mathbb{R}\to\mathbb{R}$ is continuous and injective, then it is strictly monotone. True If $f\in C[0,2]$ with $f(0)=f(2)$, then $\exists x_1,x_2\in [0,2]\ni x_1-x_2=1$ and $f(x_1)=f(x_2)$ ...
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Is the definition of continuity in analysis a particular case of topological continuity?

Take a constant function and remove an open interval from it: $$f(x)= 1, \text{if $x\in(-\infty,0]\cup[1,\infty)$ }$$ This function shouldn't be continuous because at $0$ no right limit of the ...
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Directly proving continuous differentiability

Let us say that we want to prove that a function $f: I \to \mathbb{R}$ defined on an open interval $I$ is continuously differentiable on $I$. One way to do this is to establish that $f'(x)$ exists at ...
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Discontinuous everywhere but range is an interval

Does there exist a function which is discontinuous everywhere but range set is an interval.
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1answer
31 views

Checking continuity looking whether image set is interval or not

Let $A(\neq \phi)\subseteq\mathbb{R}$. Suppose $f : A \to \mathbb{R}$ is a monotone function such that the image $f (A)$ is an interval. Then prove that $f$ is a continuous function. And if $f(A)$ is ...
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1answer
27 views

Is the function continuous that maps a vector to the coefficients of its expansion in a basis?

Let $V$ be a finite-dimensional vector space with basis $e_1, e_2, \ldots, e_n$. Consider the function $f:V\rightarrow\mathbb{R}^n$ defined such that, for each $v\in V$, $f(v)=(a_1,a_2,\ldots,a_n)$ ...
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Continuous functions satisfying $f(x)+f(2x)=0$?

I have to find all the continuous functions from $\mathbb{R}$ to $\mathbb{R}$ such that for all real $x$, $$f(x)+f(2x)=0$$ I have shown that $f(2x)=-f(x)=f(x/2)=-f(x/4)=\cdots$ etc. and I have also ...
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If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
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problem with deriving continuity equation

I am studying Aerodynamics, to be more precise, the fundamentals of Aerodynamics. The first law is the continuity equation, for which it is explained in the book that I am using. However, I wished to ...
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1answer
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$f:X\to Y \text{ is continuous} \iff f^{-1}(A^*) \subseteq (f^{-1}(A))^*$

Really struggling with exercise 9.10 from Sutherland's "Introduction to Metric and Topological Spaces". Any help would be greatly appreciated. Let $(X,t), (Y,t)$ be topological spaces, and $f: X \to ...
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Is there a topological proof that additon and multiplication are continous functions from $\mathbb R \times \mathbb R $ into $\mathbb R $?

Is there a topological proof that additon and multiplication are continous functions from $\mathbb R \times \mathbb R $ into $\mathbb R $? That is, can we prove continuity using the topological ...
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Looking for differentiable function $f:\mathbb R \to \mathbb R$ whose derivative is nowhere continuous [duplicate]

Does there exist a differentiable function $f: \mathbb R \to \mathbb R$ such that its derivative $f'$ is nowhere continuous ?
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Countable subset and monotonic function

let E be subset of R which has no isloated points(or C does not have any isolated point of E) and C be countable subset of R does there exist a monotonic function on E which is continuous only at ...
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Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
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2answers
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Counterexample for “if every continuous function on $E$ is bounded, then $E$ is compact” [closed]

Give me counter example for this false statement: "Every continuous function on the set $E$ is bounded this implies $E$ is compact".
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1answer
30 views

Finite number of jump discontinuities

Let $f : (a,b) \rightarrow \mathbb{R}$ be a monotonic function. $t \in (a,b)$ is called a jump discontinuity of $f$ if $\displaystyle \lim_{x \rightarrow t + } f(t) , \lim_{x \rightarrow t - } f(t)$ ...
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If a continuous function has one relative extremum, then it is actually an absolute extremum

The following statement makes sense intuitively, but is there a way to prove it mathematically? (This is something we make use of in applied optimization in calculus.) If $f$ is continuous on an ...
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Space $C^2(\overline{U})$ for open set $U$

Let $U$ be a bounded open domain in $\mathbb{R}^n$. Does the space $C^2(\overline{U})$ (the bar over $U$ means closure) mean the set of twice-differentiable functions $u$ such that $u, u_t, u_{x_i}$ ...
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Multivariate Weierstrass theorem?

The Weierstrass theorem states that for any continuous function $f$ of one variable there is a sequence of polynomials that uniformly converge to $f$. To my surprise, I couldn't find any reference to ...
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A random variable with distribution continuous on a parameter: Is a continuous function of such random variable continuous in the parameter?

Let $(X_n(\lambda))_{n\in\mathbb{N}}$ be a sequence of i.i.d. real continuous random variables (with density function) and assume that $P(X_n(\lambda)\le x)$ is continuous in $\lambda$. Consider the ...
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Thomae's function, doubt in continuous proof in the irrationals.

I was studying about this proof and i almost understand all of it, i just have one doubt there, the proof i found is the following; Let f be defined by; $$ \begin{align} f(x) = \begin{cases} 0 & ...
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continuity of functions on intervals

Suppose that $f : (a,b) \to \mathbb R$ is continuous. Then, there is a continuous $g : [a,b] \to \mathbb R$ such that $g(x) = f(x)$ for all $x \in (a,b)$. That is, a function defined and continuous on ...
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$f: \mathbb{R} \to \mathbb{R} $ by $f(x) =\frac 1{1+x^2}$ is uniformly continuous on $\mathbb{R} $

The definition of uniform continuity states that a function is uniformly continuous if, given any challenge $\epsilon > 0$ that there exists a response $\delta > 0$ for every value $x_1,x_2 \in ...
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Mapping the open ball to itself?

How to prove that there exists a continuous function $f:B^2 \to B^2$ without constant points? Here, $B^2$ is the unit open ball. I guess $f$ can be for example like this $f: re^{iax} \to re^{ibx} $ ...
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To show following function is discontinous

Given $f(x) = [x + 1] (\sin(1/x))$, where[.] denotes greatest integer function ; when $x\in (-1,0) \cup (0,1)$ $$f(x) = 0 , \text{ otherwise}$$ Question is to show f has discontinuity of second ...
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To show $f(x)$ is discontinuous at every point

$$f(x)=\begin{cases} 1 ,& \text {$x$ is rational} \\ 0 , & \text{$x$ is irrational}\\ \end{cases}$$ How do I show this function is discontinuous at every point. How to think about it ...
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1answer
49 views

If $b$ is a continuous function on the interval $[0,1]$, then so is its power $b^k$

If $b$ is a a continuous function on a close interval between $0$ and $1$, i.e. $b\in C([0,1])$. Let $f(b)=b^k$, $k>1$, does $f(b)$ also lies in the same interval, i.e. $f(b)\in C([0,1])$? My ...
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35 views

Continuity and diverging sequences

Let $I = (0, ∞)$ and let $f : I → \mathbb{R}$ be a continuous and bounded funciton. Show that for any real number $S$ there exists a sequence $(x_n)$ such that $\lim x_n = ∞$ and $\lim (f(x_n + S) − ...
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Holder continuity and gradient

I am trying to prove the implication of differentiability and constancy from Holder continuity. I have: $\frac{\left\lvert f(x)-f(y) \right\rvert}{x-y} \le M|x-y|^{\lambda} \implies \exists g:x ...
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4answers
71 views

Show that $A=\{x\in X\mid a\leq f(x)\leq b\:;\;a,b\in\mathbb{R}\}$ is closed if $f:X\to \mathbb R$ is continuous.

Let $X$ be a set. Suppose that $f:X\to\mathbb{R}$ is a continuous function and let $A=\{x\in X\mid a\leq f(x)\leq b\:;\;a,b\in\mathbb{R}\}$. Is $A$ closed, open, clopen or none? So I started by ...
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1answer
38 views

Will every continuous map from $S^1$ to itself have a fixed point?

Will every continuous map from $S^1$ to itself have a fixed point? I cant understand how to conclude anything from this