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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Prove uniform contininuity (probably by Lipschitz continuity)

Prove uniform continuity at $(0,\infty)$ for: $$f(x) = x + \frac{\sin (x)}{x}$$ Derivative is: $$f'(x) = \frac{x\cos (x) - \sin (x) + x^2}{x^2}$$ so, taking the limit at $\infty$ I got the value of ...
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1answer
73 views

Continuity of $h(x)=f(x) \cdot g(x)$

$h(x)=f(x) \cdot g(x)$ I want to check whether this function is continuous in its domain $\mathbb{R}$ or not. definition by cases: $f(x)$ and $g(x)$ are both continuous $\Rightarrow f(x) \cdot ...
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3answers
130 views

Continuity of $f(x)=\max\{x,0\}$

$$f(x)=\max\{x,0\}$$ I want to check whether this function is continuous in its domain $\mathbb{R}$ or not, but unfortunately I have no idea how to start.
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2answers
144 views

Example of continuous function that isn't uniformly continuous and isn't 1/x

I understand that in an open interval the only functions that are continuous but not uniformly are functions whose limits are singularities. But when we have a function $f:H\rightarrow\mathbb{R}$ and ...
2
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1answer
77 views

Proving and disproving $\exists b,c\in \mathbb R$ such that $f(x)=\frac a2x^2+bx+c $

Let $f:I\to\mathbb R$ where $I$ is an interval, $f''(x)=a \ \ \forall x\in I$. Prove that there exsits such numbers $b,c\in \mathbb R$ such that: $f(x)=\frac a2x^2+bx+c ,\ \forall x\in I$. ...
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1answer
46 views

Proving this Taylor-esque expansion for a $C^2$ function vanishing at 0 and 1

I am trying to prove the following (which I think is true!): if $f:[0,1]\rightarrow \mathbb{R}$ is twice continuously differentiable and $f(0)=0=f(1)$, then for every $x \in (0,1)$ there exists $\xi ...
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2answers
30 views

Continuity of the function

Is the sequence a continuous function on the set of natural numbers? My book on complex numbers insists that for the function to be continuous, the limit at a point must exist, which, of course, makes ...
2
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1answer
150 views

$x_n$ convergence to $x$ implies $f_n(x_n)$ convergence to $f(x)$. prove that $f$ is continuous

Let $f$ and $f_n$ be functions from $\mathbb{R} \rightarrow \mathbb{R}$ Assume that $f_n (x_n) \rightarrow f (x)$ as $n\rightarrow \infty$ whenever $x_n \rightarrow x$. Prove that $f$ is ...
3
votes
3answers
174 views

Prove $x^{2 \over 3} \ln(x)$ is uniformly continious

Prove $x^{2 \over 3} \ln(x)$ is uniformly continuous in $(1,\infty)$ To my understanding I need to show the derivative is bounded. That will prove uniform continuity. The derivative is: $$ ...
0
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3answers
57 views

Compact subsets of a metric space

I am trying to to prove that f: X --> Y is continuous on X if and only if f is continuous on every compact subset of X. X and Y are metric spaces. How do I show that every point of X belongs to some ...
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1answer
159 views

Redefine Function to Solve Discontinuity

$$ f(x) = \frac{6x^2-5x-4}{2x^2+x} $$ $f(x)$ is discontinuous at $x = -1/2$ Redefine $f(-1/2)$ so that the discontinuity can be removed
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1answer
195 views

Prove $f(x) = \sqrt {\ln x} \ln (\ln x)$ is uniformly continuous.

Let $f(x) = \sqrt {\ln x} \ln (\ln x)$ Prove $f(x)$ is uniformly continuous. I'd be glad to get hint/guidance. I tried to follow the definition of uniformly continuous, but got stuck in the very ...
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1answer
48 views

Relationship between f(X) and f(closure of X)

I am trying to prove if f is continuous and closed ("closed" means the image of any closed subset of the domain is closed) then f(closure of X) equals the closure of f(X). I was able to prove that if ...
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1answer
22 views

Basic Continuity issue.

F is differentiable . We know that : Lim F(x^2) = F(0) when x^2 -> 0 How can we show in an easy way that : Lim F(x^2) = F(0) when x->0 Can we derive this directly from continuity, no ...
3
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1answer
56 views

is continuity preserved under Expectation?

Let's say I have a random function $X(t)$ that is continuous in $t$, almost surely. Is it true that $$\mathbb E(X(t_1)) = \mathbb E\left(\lim_{t\to t_1} X(t)\right)?$$ This seems incorrect to me ...
2
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1answer
95 views

Difference between expressions regarding Lipschitz continuity

Let $f:\mathcal{I}\times \mathcal{X} \to\mathbb{R}$ be an arbitrary function, e.g., $f(t,x)=t^2+x$. What are the differences between the following locally Lipschitz continuity definitions: "$f$ is ...
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1answer
46 views

Difference between expressions regarding continuity

Let $f:\mathbb{R_{\geq 0}}\times \mathbb{R}^n\to\mathbb{R^n}$ be an arbitrary function, e.g., with $n=1$, $f(t,x) = t^2+x.$ What is the difference among the following expressions: "$f$ is continuous ...
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1answer
16 views

How do the Fourier Transform of sampling and the Frequency-domain convolution match?

The Fourier Transform(FT) is $X(\upsilon) = \int_{-\infty}^{\infty}x(t)e^{-2{\pi}i{\upsilon}t}dt$. The impulse train is $\delta_1(x)=\sum\limits_{k=-\infty}^{\infty}\delta(x-k)$, and its FT is ...
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2answers
40 views

Continues function so lim(x)=lim(x^2) x->0?

Let $F$ be a continuous function on $\mathbb{R}$. Can we derive from here that $\lim_{x \to 0} F(x)= \lim_{x \to 0} F(x^2)$? I think that it's true because of Heine But I can't find a way to prove it ...
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2answers
60 views

Convergence of $f(p_n)$ insufficient to show continuity?

I came across a problem that asked one to assume $f: M\to N$ is a function from a metric space to another, and that if $(p_n)$ in $M$ converges then $f(p_n)$ in $N$ converges. It asked that one show ...
4
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1answer
115 views

Proving continuity of exp(x)

Well, my teacher went through a method of proving continuity of $\exp(x)$ which I don't like, so I tried to go about it a different way: We have proved the following (which I use) $\exp(x+y) = ...
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9answers
790 views

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
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Continuity in multivariable calculus

I want to find out the points, where the function $f(x,y)=\dfrac{xy}{x-y}$ if $x\neq y$ and $f(x,y)=0$ otherwise, is continuous. I have shown that at all the points $(x,y)$, where $x\neq y$, $f$ is ...
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97 views

(…) Show that $f(x) ≤ f(b)$ for all $x \in [a,b]$ and that $f(a) = f(b)$

Can someone help me with this proof? Let $a,b \in\mathbb{R}$ with $a < b$. Furthermore let $f: [a,b] \rightarrow \mathbb{R}$ be continuous with the following conditions: For $\forall x ...
2
votes
3answers
197 views

Uniform continuity of continuous function on a subset

Assume that $f: \mathbb R \rightarrow \mathbb R$ is continuous on the compact set $A$. Does for any $\varepsilon >0$ exist a $\delta >0$, such that $$ \lvert\, f(x)-f(y)\rvert<\varepsilon ...
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1answer
90 views

Constructing explicit lift of a circle homeomorphism

Studying a book by Luis Barreira in the Universitext Collection -- Dynamical Systems: an Introduction -- I'm told that given $f: S^{1} \to S^{1}$ homeomorphism, it's always possible to construct a ...
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2answers
372 views

Showing that the function is continuous but not differentiable

Let $$ f(x,y) = \begin{cases} \dfrac{xy}{\sqrt{x^2+y^2}} & \text{if $(x,y)\neq(0,0)$ } \\[2ex] 0 & \text{if $(x,y)=(0,0)$ } \\ \end{cases} $$ Show that this function is continuous but not ...
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238 views

Is $\log (1 + {x^2})$ uniformly continuous on $[0,\infty)$? [duplicate]

Is $\log (1 + {x^2})$ uniformly continuous? Here is my attempt: Let $\forall\left| {x - y} \right| < \delta$: $\left| {\log (1 + {x^2}) - \log (1 + {y^2})} \right| = \left| {\log (\frac{{1 + ...
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2answers
90 views

Prove/Disprove $f(x) = x + \frac{x}{{x + 1}}$ is uniformly continuous at $\forall x,y \in [0,\infty )$ [duplicate]

Prove/Disprove $f(x) = x + \frac{x}{{x + 1}}$ is uniformly continuous at $\forall x,y \in [0,\infty )$ This is my trial: $$\forall \varepsilon > 0\exists \delta > 0.\forall x,y \in ...
2
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3answers
185 views

Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$

Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$ Going from the epsilon delta definition we get: $$\forall x,y>1,\text{WLOG}:x>y \ ,\ \forall\epsilon>0,\exists\delta>0 ...
4
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1answer
90 views

Continuity of double centralizers in Banach algebras

I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let $A$ be a ...
2
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1answer
80 views

Uniform continuity - Please check my work

Iv'e got some questions involving uniform continuity of functions and its properties. I would like that someone will check my work and maybe help with correcting flaws. Let $A \subset R$ and let ...
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3answers
827 views

Fixed-Point Theorem Proof

Merry Christmas everybody. Let $a,b\in\mathbb{R}$ and $a<b$. Prove that: If $f: [a,b] \rightarrow [a,b]$ is continuous, then there is a fixed-point in $f$. So basically, if f is continous I ...
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2answers
54 views

Proving discontinutiy with epsilons and deltas.

I never actually had to show a function was discontinuous before, but trying to.. stumped me. I tried with the funtion $$f(x)= \begin{cases} x &\mbox{if } x\leq 1\\ x+1 &\mbox{if } x > ...
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1answer
62 views

How can I define a “gradient discontinuous function”?

I am writing a report and need to know how I can define the "kink" in |x|. The function technically adheres to the definition of continuity, and the left and right limits appear to agree here... I ...
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1answer
41 views

What's behind the function $g(x)=\operatorname{inf}\{f(p)+d(x,p):p\in X\}$?

In several books on measure theory, I have seen the following problem: Suppose $(X,d)$ is a metric space, on which $f$ is a nonnegative lower semicontinuous function. Show that $f$ is the ...
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0answers
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What do they mean by this: “the definition of product topo shows that this mapping is continuous”

Let $\Bbb{Z}_2$ be the $2$-adic integers. There's a bijection $\psi : \Bbb{Z}_2 \to C$, the Cantor set, defined by $\psi (\sum_{i \geq 0} a_i p^i) = \sum_{i \geq 0} \dfrac{2a_i}{3^{i+1}}$. The text ...
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2answers
103 views

Differentiability of a certain piecewise function

Consider the function $$ f(x)=\begin{cases} x & \textrm{if } x \textrm{ is rational} \\ -x & \textrm{if } x \textrm{ is irrational} \end{cases} $$ It is well-known that $f(x)$ is continuous at ...
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213 views

Continuous mapping $f: [0,1]\rightarrow (0,1)$ CSIR December $2013$

Question is : Suppose $f: [0,1]\rightarrow (0,1)$ is Continuous then which of the following is NOT true.. $F\subseteq[0,1]$ is closed set implies $f(F)$ is closed in $\mathbb{R}$ If $f(0)<f(1)$ ...
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1answer
60 views

To check whether any of the following is an ideal in the ring of continuous real valued functions

Let $C(\mathbb R)$ denote the ring of all continuous real valued functions on $\mathbb R$, with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal ...
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1answer
57 views

Iterating a real continuous injective function having no fixed points.

Let $f: \mathbb R \rightarrow\mathbb R$ be a continuous injective function. If $f(x)≠x ,\forall x∈\mathbb R$ and there exists a positive integer $n$ such that $f^n(x)=x , \forall x∈\mathbb R$ , then ...
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1answer
139 views

Proving that $f(x)\geq 0$ on $[0,1)$ when $f(x)$ is continuous and when the Darboux/Riemann integral of $f(x)$ is greater than 0.

Suppose that $f(x)$ is continuous and defined on $[0,1)$. Also, suppose that the Riemann/Darboux integral $\int_a^b f(x)\geq 0$ on $[0,1)$ for any partition . Show that $f(x)\geq 0$ for all $x\in ...
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1answer
382 views

Proof of a claim on a continuous function in [0,1] [duplicate]

This question has given me a huge headache, and I can safely say I hate everything about it. I need to prove, that for a continuous function in [0,1],which has the property that for every x in the ...
2
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2answers
180 views

prove that the following equation has exactly 2 solutions [duplicate]

Let $ a_1,a_2,a_3 >0$ $\lambda_1\lt \lambda_2 \lt \lambda_3 \in \mathbb{R}$ prove that the following equation has exactly 2 solutions $\dfrac{a_1}{x-\lambda_1} + ...
3
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2answers
105 views

Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $

Let $f$ a continous function defined in the interval $[0,1]$. Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $ I tried to use Heine–Cantor theorem ...
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725 views

Which functions satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all ...
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1answer
44 views

Continuity problem involving piecewise function and glue points

Determine all values of the constant a such that the function f(x) below is continuous at the glue point. $$f(x)= \{ \frac{ax}{tanx}, x>0; a^2-2,x\leqslant0$$ I tried to do many things like ...
0
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1answer
53 views

Different types of continuity in $\ell^2$

Consider the following functional $J$ on $\ell^2$ which for $x = \{x_n\}$ is defined by $$J(x) = \sum_{n=1}^{\infty}n^{1/n}x_{n}^{2}.$$ Is $J$ continuous? Is $J$ lower semi-continuous? Is $J$ ...
0
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1answer
77 views

Let $f$ be a continuous function on $[0,2]$ such that $f(0)=f(2)$, prove the following statement.

I've recently come across this exercise for which I currently have no clue on how to solve. Basically, it says that given a function $f$ that is continuous on $[0,2]$ such that $f(0)=f(2)$, you have ...
0
votes
3answers
130 views

to prove that $f(x)=f(0)$

Let, $f:(-1,1)\rightarrow\mathbb{R}$ be a function continuous at $x=0$ and given that $f(x)=f(x^2)$ for all $x\in(-1,1)$. Prove that, $f(x)=f(0)$ $\forall x\in(-1,1)$. Ok. First give me some hint. ...