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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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 ...
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50 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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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 $\delta=\frac{\epsilon-|g(2)|}{M}$ ...
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3answers
85 views

Prove $A=\{x\in \mathbb{R}|f(x)=x\}$ is closed subset of $\mathbb{R}$

Given $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is continuous function. Prove that $A=\{x\in \mathbb{R}|f(x)=x\}$ is closed subset of $\mathbb{R}$. I totally do not have idea on how to start the ...
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2answers
182 views

If two continuous maps into a Hausdorff space agree on a dense subset, they are identically equal [duplicate]

Let $f, g : X \to Y$ be continuous functions. Assume that $Y$ is Hausdorff and that there exists a dense subset $D$ of $X$ such that $f(x) = g(x)$ for all $x \in D$. Prove that $f(x) = g(x)$ for all ...
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39 views

Uniform continuity of scalar multiplication in topological vector spaces

If $X$ and $Y$ are topological vector spaces over $\mathbb R$, then a map $f:X\to Y$ is called uniformly continuous if for each neighborhood $V\subseteq Y$ of $0\in Y$, there exists a neighborhood ...
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34 views

Uniform continuity of the antiderivative

We know that if $f:\mathbb{R}\to\mathbb{R}$ is a function such that $$\sup_{x\in\mathbb{R}}|f(y)|<\infty,$$ then the function $g(x)=\int_0^xf(y)dy$ is uniformly continuous. I am just wondering ...
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75 views

Monotonically increasing $f$ .

Let $f$ be a monotonically incresing function from $[0, 1]$ into $[0, 1]$. Which of the following statements is/are true? $(1)$ $f$ must be continuous at all but finitely many points in $[0, ...
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156 views

A continuous function on $[0,1]$ not of bounded variation

I'm looking for a continuous function $f$ defined on the compact interval $[0,1]$ which is not of bounded variation. I think such function might exist. Any idea? Of course the function $f$ such ...
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40 views

$A:=\left\{y\in\mathbb{R}:\mu\left(f^{-1}(\left\{y\right\})\right)>0\right\}$ is countable, if $\mu$ is a finite measure and $f$ has compact support

Let $E$ be a metric space $\mathcal{E}:=\mathcal{B}(E)$ be the Borel algebra on $E$ $\mu:\mathcal{E}\to [0,1]$ be a measure $f:E\to\mathbb{R}$ be measurable and have compact support Assume that ...
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1answer
92 views

Example of two closed continuous functions whose “product” is not closed

Let $X,Y,W,Z$ be topological spaces, and let $f: X \longrightarrow Y$, $g: W \longrightarrow Z$ be closed functions. Let $f \times g: X \times W \longrightarrow Y \times Z$ be such that $f \times ...
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133 views

A continuous function such that $f(x)=(f(x))^2$ for all $x$ is constant

Let $f(x)=(f(x))^2$ that is continuous for every $x \in\mathbb R$. Prove using the intermediate value theorem that this function is constant. I noticed that the $f(x)$ could only be equal ...
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1answer
52 views

Can $x= x_0$ in the process of $x \to x_0$?

In the process of $ x→x_0 $,can $x$ get the value of $x_0$ ? i.e. can $x= x_0$ here ?
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1answer
24 views

Prove that a $f(c)=c$ given an interval where the function is continuous over

If $f(x)$ is continuous over the interval $[a,b]$ $a,b \in R$ $ a<b$ such that $f(a), f(b)$ also belong to the interval $[a,b]$ Prove that there exists some value $c$ in $[a,b]$ such that ...
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1answer
38 views

For which constants $a$ function is continuous

Let $f(x)= \lfloor x \rfloor \cdot\cos{(a\cdot x)}$ where $x\in \mathbb{R}$. Find for which real constants $a$ function is continuous. We know function $ \lfloor x \rfloor$ is continuous apart from ...
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1answer
17 views

Prove that a function has 2 solution and find one solution using the bisection method

$ln(x^2+2x+\frac{1}{2})=x$ Prove that this equation has 2 solution over the interval $[0,10]$ Find the two first digits of one of the solution using the bisection method. I started with defining ...
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3answers
61 views

Prove that a polynomial function has a root

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ A polynomial function such that $n$ is odd positive integer and $a_n\ne0$ Prove that this function has a root. I tried and eventually come to the ...
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4answers
168 views

Question about Intermediate Value Theorem

In the solution, it says that $f(a)\ge a$ and $f(b)\le b$ but it do not seem obvious for me. If I am just given that a $f:[a,b]\to[a,b]$, how do I know is this function increasing, decreasing or ...
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1answer
87 views

Why do we need to assume continuity in the proof of the chain-rule?

Look at this proof: If $f$ is differentiable at $x$, then it must be continuous there too? Does he then need in the hypothesis that $f$ need to be continuous in the entire interval? What if he just ...
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1answer
50 views

Continuous extension on compact set in $\mathbb{R}^n$

I'm an undergrad student reading through Deimling's Nonlinear Functional Analysis and have come across the following proposition. Let $A\subset\mathbb{R}^n$ be compact and $f:A\to\mathbb{R}^n$ be a ...
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41 views

Lipschitz continuity of a generalized Rayleigh quotient

I am thinking about the Lipschitz continuity of a generalized Rayleigh quotient: $f(x)=\frac{x^\top Ax}{x^\top Bx}$ with the constraint $||x||\geq c$, where both $A$ and $B$ are positive definite ...
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50 views

Can I change the proof this way?

In the proof I think he takes intersections of forward images. I am not that comfortable with this because I remember there was some problems with that. So I change it to inverse images. Is the proof ...
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1answer
38 views

Continuous function changing sign on Cantor set

Let $C$ be the Cantor set. I can easily define a continuous function on $[0,1]$ whose set of zeros is exactly $C$, i.e. $x\mapsto d(x,C)$. In addition to being 0 on $C$, I'd like to build one that ...
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1answer
25 views

Continuity of a function in a mixed discrete-connected domain

Consider the following function $f: \mathbb{R} \times \left\{0,1\right\} \rightarrow \mathbb{R}$ $$ f(x,y)=\begin{cases}x, & (y=0) \\ k, & (y=1) \end{cases} $$ where $k$ is some real ...
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1answer
43 views

Why is $\varphi\colon A^G\to A$ continuous?

Let $G$ be a group and $e_G$ its neutral element. Moreover, let $A$ be a finite set. $A$ is equipped with the discrete toplogy, $A^G$ with the product topology. Let $\tau\colon A^G\to A^G$ be ...
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122 views

Is a constant function between topological spaces continuous?

Let $T\colon X\to Y$ be constant, where $(X,\tau_1)$ and $(Y,\tau_2)$ are topological spaces. Maybe a silly question, but is then $T$ continuous? It is to show that for $O\in \tau_2$ I have that ...
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1answer
81 views

Continuous Nowhere Differentiable Function [closed]

Define a function $\,f:\mathbb{R}\rightarrow \mathbb{R}_{+}$ by: $$ f(x)=\left|x-2\,\left \lfloor \frac{x+1}{2}\right \rfloor \right|. $$ Here are some known properties about the function $f$: ...
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174 views

Show that $f(x) = x\cos^3(x)$ is not uniformly continuous on $\mathbb{R}$

Show that $f(x) = x\cos^3(x)$ is not uniformly continuous on $\mathbb{R}$. I tried $x_n = \pi/2 + n\pi$ and $y_n = \pi/2 + n\pi + 1/(n\pi)$. Since $\cos^3(x) = \frac14 (\cos(3x)+3\cos x)$, ...
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eventually constant maps

Let $f:[0,1]\to [0,1]$ be a continuous function with a unique fixed point $x_{0}$ Assume that $\forall x\in [0,1], \exists n\in \mathbb{N}$ such that $f^{n}(x)=x_{0}$. Does this implies ...
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1answer
43 views

How to prove this map is continuous

Let $Y=\mathbb{R}^2 \setminus\left\{\begin{bmatrix}0\\ 0 \end{bmatrix}\right\}$ and $I=\left[0,1\right]$, both with the subspace topologies of the Euclidean ones. Define a map $F:Y\times I\to Y$ by ...
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1answer
34 views

Looking for another special kind of injective function

Relating to this Looking for a special kind of injective function Does there exist an injective function $f:\mathbb R→\mathbb R$ such that for every $c∈\mathbb R$ , there is a real sequence ...
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$f(0)=0$ and $\lvert\,f^\prime (x)\rvert\leq K\lvert\,f(x)\rvert,$ imply that $f\equiv 0$.

I'll start with the precise statement of the problem: Suppose that $f:[0,b]\to\mathbb{R}$ is differentiable, $\,f(0)=0$, and that there exists a real number $K\geq 0$ such that ...
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1answer
45 views

Equivalent condition for continuity of a function

Let $g: [0,+\infty) \rightarrow \mathbb{R}$ be a continuous function and let $f: [0,+\infty) \rightarrow \mathbb{R}$ be defined by \begin{equation} f(t) = \inf \{ s \geq 0 \,|\, g(s) > t\}. ...
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2answers
31 views

Explanation of Proof for (Dis)continuity of Thomae's Function

Can someone explain the proof for the continuity at irrationals but discontinuity at rationals for Thomae's function? More specifically, why if x is rational with $x = \frac{p}{q}$, and we choose ...
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3answers
83 views

Stone-Weierstrass: Examples

By Stone-Weierstrass one has: $f\in\mathcal{C}(K):\quad p_n\to f$ Now, for analytic functions this is just Taylor: $$f\in\mathcal{C}^\omega([a,b]):\quad f(x)=\sum_{k=0}^\infty a_kx^k$$ But, how does ...
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1answer
56 views

Application of Rolle's Theorem and differentiation

Suppose $f: \mathbb{R}\rightarrow \mathbb{R}$ is differentiable with $f(0)=f(1)=0$ and $\{x:f'(x)=0\}\subset \{x:f(x)=0\}$. Show that $f(x)=0$ for all $x\in [0,1]$. My Work: By Rolle's Theorem ...
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1answer
32 views

differentiabilty implies continuity (analysis)

Is my proof correct? We need to show that if $f$ is differentiable at $x_o$, then it is continuous at $x_o$ i. e. $$\forall \epsilon >0, \exists \delta >0 \text{ s.t. } ...
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4answers
59 views

Prove that a function is continuous for every $x \in R$

Prove that the function: $$ f(x)=\frac{\sqrt{x^2-x+1}}{|\sin(x)-4|-2} $$ is defined for every $x \in R$ and continuous in every $x \in R$, So I said that in order for this function to be defined we ...
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0answers
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Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...
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1answer
35 views

Check my answer - simple laplace transform of piecewise continuous function.

I'd just like to check that I got the idea right, first exercise im doing in laplace transforms and am a bit clueless. We are given $f(t)=0$ if $0<t<2$ and $f(t)=t$ if $t>2$. We are asked to ...
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3answers
106 views

Multiple choice question about limits and continuity? (Or, $\tan x$ is continuous?!)

I'm doing a test about limits and continuity and got these two wrong. $\mathbf{Q1}$: The function $f(x) = \tan x$: $\hspace{1em}\mathtt{a)}$ is continuous $\hspace{1em}\mathtt{b)}$ is ...
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1answer
40 views

Looking for a special kind of injective function

Does there exist an injective function $f:\mathbb R \to \mathbb R$ such that for every $c \in \mathbb R$ , there is a real sequence $(x_n)$ such that $\lim\big(f(x_n)\big)=c$ but $f$ is neither ...
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0answers
36 views

Looking for a partial converse of Rolle's theorem

Let $f: [a,b] \to \mathbb R $ be a continuous function differentiable in $(a,b)$ such that $f(b)=0$ and for some $c \in (a,b) , f'(c)=0$ ; then under what additional conditions can we conclude that ...
4
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1answer
54 views

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

Is there any result in fixed point theory which will give the existence of a fixed point for a continuous function defined on a non-compact, closed and bounded convex set?
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2answers
27 views

Find the continuous function such that the Riemann integrable is the same

Find all functions $f$ such that $f$ is continuous on $[0,1]$ and $\int_0^x f(t) dt = \int_x^1 f(t) dt$ for every x $\in (0,1)$ I can't think of any function that would satisfy this property! ...
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0answers
69 views

How show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$. Issue: I know how to prove this via the epsilon-delta way. I ...
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1answer
28 views

Limit of arc-length of a curve

Let $L(f)$ denote the length of a curve $f$, if $f = \lim\limits_{n\to\infty} f_n$ then do we necessarily have that $L(f) = \lim\limits_{n\to\infty} L(f_n)$? I assume that we will have some continuity ...
3
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2answers
101 views

Limit of a continuous function with a parameter

Let $f(x,\alpha)$ be continuous function on $S=(0,1]\times[0,1]$. Suppose that for every segment $[\alpha,\alpha+\Delta\alpha]\in[0,1]$ there exists $x_0=x_0(\Delta \alpha)$ s.t. for $0<x<x_0$ ...
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2answers
69 views

How to find a continuous function that demonstrates that the set $\{(x,y):y>x\}$ is open?

Consider the set of points $U$ in $\Bbb{R}^2$ that lie above the line $y = x$, i.e. points $(a,b)$ such that $b>a$. Prove that $U$ is open and connected. The method that is recommended is showing ...