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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Function continuous at end-points

If we have a function $f$ that is absolutely continuous on $(-1,1)$ (and also the derivative of $f$ is absolutely continuous on $(-1,1)$) and we have that for the derivative the following limits ...
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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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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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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
25 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
54 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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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
17 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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69 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
33 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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33 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 ...
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1answer
48 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
25 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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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
24 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 ...
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72 views
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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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61 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 ...
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1answer
89 views

Determinant: Continuity

Reference Build-up on: Determinant: Definition Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its determinant $\det:\mathcal{L}(V)\to\mathbb{C}$. Introduce a norm ...
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Does there exist a continous function $f(t)$ on $[0,1]$ for which $\int_0^1 t^3 f(t) dt = 0$?

Does there exist a continous function $f(t)$ on $[0,1]$ for which $\int_0^1 t^3 f(t) dt = 0$? Or can you provide a proof otherwise?
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1answer
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Prove $f:\mathbb{N} \rightarrow \mathbb{R}$ is continuous using the definition of sequential continuity

The definition of sequential continuity is that $x_n \rightarrow x \implies f(x_n) \rightarrow f(x)$. If the terms of the sequence $\{x_n\}$ are only natural numbers, I know that for all $\epsilon ...
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Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
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Is this a criterion for continuity?

Given a topological space $(X,\tau)$ and the product space $(X^2,\tau_2)$. Define the diagonal $\Delta X^2=\{(x,x)\,|\,x\in X\}$ and a set $\mathbf S_\tau=\{\mathcal A\in\tau_2|\Delta ...
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let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true? (a). If $g$ is continuous, then $f\circ g$ is continuous. ...
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Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
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2answers
46 views

Prove that a continuous inverse-transformation of $f: [0,1) \cup \{ 2 \} \to [0,1]$ exists

I am having this transformation $f: [0,1) \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ I've already proved that it is continuous. Question: Is ...
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Existence of continuous functions $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ ; and what if $(0,1)$ replaced by $[0,1)$ ?

Does there exist continuous functios $f,g:(0,1) \to (0,1)$ such that $f\big((0,1)\big)=(0,1)$ \ $g\big((0,1)\big)$ ? The problem I am having is that since $(0,1)$ is not compact I am not able to tell ...
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How to proove that a bijective transformation is NOT continous

I am having this transformation $f: \mathbb R \to \mathbb R$ $$f(x) = \begin{cases} x & x \in \mathbb R \setminus \mathbb Q \\x+1 & x \in \mathbb Q \end{cases}$$ I've already prooved ...
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Teacher Challenge - multiple parts

This was his challenge: "I would like you to consider the function $x^{r+\alpha}$, $r$ is an integer, $\alpha$ is a real number between $0$ and $1$. Differentiate it until you get a singularity ...
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A basic doubt on upper semi-continuity of set-valued maps

Upper Semi-Continuity for set valued maps have two definitions $h:\Bbb R^d \to 2^{\Bbb R^d}$ is upper semi-continuous if 1) Sequential definition : $x_n \to x$, $y_n \to y$ and $y_n \in h(x_n)$ ...
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Requirements for existence Lebesgue-Stieltjes measure corresponding to distribution function in $\mathbb{R}^n$

I am going through Ash's book "Probability and Measure Theory". It says that: We know that a distribution function of $\mathbb{R}$ determines a corresponding Lebesgue-Stieltjes measure. This is true ...
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Prove that $f\left(x\right)=\sin\left(x\right)$ is Continuous.

The function $f\left(x\right)=\sin\left(x\right)$ is obviously continuous. But how would you prove this using the $\delta,\varepsilon$ definition of continuity? So given $x\in\mathbb{R}$ and ...
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Is $f: [0,1[ \cup \{ 2 \} \to [0,1]$ continuous?

I am having this transformation $f: [0,1[ \cup \{ 2 \} \to [0,1]$ $$f(x) = \begin{cases} x & x \neq 2 \\1 & x = 2 \end{cases}$$ How can I prove that this transformation is continuous or ...
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Showing points of continuity of a function f(x) that takes the value 1/n whenever x belong to a sequence {An} and is zero elsewhere.

I am given a sequence $(An), n=1,2,3,...$ which consists of distinct numbers, which converges to $3$ as $n$ tends to infinity, but none of its terms are equal to $3$. Then I am given a function $f(x) ...
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Is function $f$ also uniformly continuous?

I've been thinking on the following problem lately: Let $(X,d)$ be a metric space and $f_1,f_2,...,f_n: X \rightarrow \mathbb{R}$ and $f(x) = \max\{f_1(x),f_2(x),...,f_n(x) \}$,$x\in X$ If the ...
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$X\subset \mathbb R$ and $f,g:X\to \mathbb R$ be continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)=X$ [closed]

Let $X\subset \mathbb R$ and $f,g:X\to \mathbb R$ be continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)=X$, Which of the following sets can not be equal to $X? $ A. ...
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Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
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Proving uniform continuity and uniform discontinuity

Could someone please explain to me how to show uniform continuity and not uniformly continuous for the following: $f(x) = \frac{1}{x^2}$ for $A = [1, \infty)$ show uniform continuity $f(x) = ...
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Need to show the following function is uniformly continuous on R

Could you please tell me how I am supposed to show that $f(x) = \dfrac{1}{(1+x^2)}$ is uniformly continuous in $\mathbb{R}$. I did some pre-calculation and found that $|f(x) - f(u)| < \epsilon$ if ...
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1answer
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Show that $f(z):=\sum a_n (z-z_0)^n$ is continuous whenever $z$ is in disk of convergence.

Consider a power series $\sum a_n(z-z_0)^n$, and assume it has radius of convergence $r$. Then we know that $\forall z\in(z_0 -r,z_0 +r)$, this power series converges absolutely by root test. Thus we ...
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How to show that a real continous function with image in the rationals is constant?

Can someone please explain to me how I am supposed to approach this question: If $f: [0,1] \to \mathbb{ R}$ is continuous, and has only rational values, then $f$ must be a constant.
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How to approach this problem?

Could someone please explain to me how I am supposed to approach and prove the following problem: Let $I= [a,b]$ and $f:I \to \mathbb{R}$ be a continuous function on $I$ such that for each $x \in I$, ...
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If a continuous function is positive on a closed interval $I$, there exists a positive number $\alpha$ such that $f(x) > \alpha$ for all $x\in I$

Could someone please explain to me how I am supposed to how I am supposed to approach this question: Let $I = [a,b]$ and $f:I\mapsto \Bbb R$ be a continuous function on $I$ such that $\forall x\in ...
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Defining Topological Continuity

I have seen this definition many times: Topological Continuity: A function $f:X\rightarrow Y$ is continuous if for all open sets $U \subseteq Y$, the preimage $f^{-1}(U)$ is open in $X$. I don't ...
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Need help in the continuity question [duplicate]

could someone please explain to me the following question: Let $f,g$ be continuous functions from $\mathbb{R}$ to $\mathbb{R}$ and suppose that $f(r) = g(r)$ for all $r \in \mathbb{Q}$. Is it true ...
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Find two functions $f$ and $g$ such that they are both discontinuous at $c$, however, $f+g$ and $f\cdot g$ are both continuous at $c$

Could someone please explain to me how to approach these kinds of question and also what is the answer to the following question? Give an example of a function $f$ and $g$ such that they are both ...
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38 views

No continuous transformation $f([a,b])= ]a,b[$

$ a,b\in\mathbb R$ with $a<b $. Now I want to show that there is NO continuous transformation $f: [a,b] \to \mathbb R $ with $f([a,b])= ]a,b[$ How can I proove that this transformation don't ...
5
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2answers
141 views

$f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?

Let $f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$? A. $\{0\}$. B. $(0,1)$. C.$[0,1)$. D.$[0,1]$. ...
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35 views

Continuity of normal curvature

I want to show that the normal curvature is a continuous function. At first, here is the definition of normal curvature at point $p \in M \subset \mathbb{R}^3$ in direction $\mathbf{u} \in T_{p}M$: ...
3
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1answer
54 views

Theorem 4.22 from baby Rudin: continuity and connectedness

I have some parts that I don't understand from the given proof. The theorem is: If $f$ is a continuous mapping of a metric space $X$ in to a metric space $Y$, and if $E$ is a connected subset of $X$, ...
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1answer
36 views

Extreme point of unit balls, over $\mathbb C$

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $\mathcal{C}[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...