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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Subtle Analysis Problem

Suppose you have a function $f \colon A \to \mathbf {R} $ and $ (a - \delta', a + \delta') \subseteq A$ for some $\delta' > 0$. Suppose also that $f$ is continuous at $a$. How do you prove that the ...
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Proving $f: A \to R$ is continuous at $a \in A$ knowing $(a − \delta', a + \delta') \subset A$ for some $\delta' > 0$

I've been working on this question for a while now and I can't seem to figure it out. Suppose $f: A \to R$ is a function and $A$ contains an interval $(a − \delta', a + \delta')$ for some $\delta' ...
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24 views

Proof of continuity via Sequence Criterion?

We are to prove that $f(x) = x$ if $x$ is rational, and $f(x) = 1 - x$ if $x$ is irrational is discontinuous for all $x$ on the interval $[0,1]$ except at $x = 1/2$. So, I've broken the proof into two ...
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Is the bijectivity of a function equivalent to monotony and continuity?

My high-school math professor told us that in order for a function $ f $ to have a reverse it must be monotonic and continuous, but I always thought that necessary and sufficient condition for a ...
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A connected path between shapes

This is a follow-up to this question: A continuous path between shapes . Let $A$ and $B$ be two measureable, bounded, connected subsets of $\mathbb{R}^2$ such that $A\subseteq B$. Does there exist a ...
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Increasing function non-continuous on points of sequence - construction

How to construct strictly increasing function $f$, non-continuous on points of countable sequence of numbers $a_n$?
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Angel function and continuity

I have the function $w:\mathbb{R}^2\backslash\{0\}\rightarrow\mathbb{R}$ given by $\cos(w)=\frac{x_1}{||x||_2}\text{ and }\sin(w)=\frac{x_2}{||x||_2}$ after some manipulation I got ...
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Norm in $C(X,\Bbb{R})$

Let $X\subset\Bbb{R}$ a compact set and $f\in C(X,\Bbb{R})$. Define $$\|f\|_{\infty}=\sup A_f$$ with $A_f=\{|f(x)|\in \Bbb{R};x\in X\}$. Then $\|f\|_{\infty}=|f(x_0)|$, for some $x_0 \in X$, since ...
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continuity of a function

$$f(x) = \begin{cases}(1-\cos x)/x & x \neq 0\\0& x=0\end{cases}$$ I am asked to prove if it is continuous at $x_1=0$ $$|f(x)−f(c)|<\varepsilon$$ Since $$1-\cos(x)=2\sin^2(x/2)$$ ...
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37 views

$f:[a,b]\to [c,d]$ be a monotone, bijective map, $f^{-1}$ is continuous?

I am sure that $f$ must be continuous.My intuition says $f^{-1}$ need not be continuous but I have no counter example. $2,3,4$ are surely false. Could any one help me to solve this problem?
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Continuity of a function at $0$

A similar has been asked before, but it was confusing. Please help me with it. I need a general method of dealing with such problems I need to show that the following function is continuous at $0$. ...
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37 views

Is a continuous function >0 and defined on an open interval bounded by a constant?

If g is continuous on (a,b) and g(x) > 0 for all x ∈ (a,b), then there is some constant M > 0 such that g(x) ≥ M for all x ∈ (a,b). True or False? I think this is false since g is defined on an open ...
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37 views

How do we prove the continuity of the exponential function restricted to $\mathbb{Q}$?

Let $M$ be a natural number and, for $p/q\in \mathbb{Q}$, define $M^{p/q}$ as $\sqrt[q]{M^p}$ (forget about $a^x$ when $x$ is not rational). Prove that $f:\mathbb{Q}\to \mathbb{R}$ is continuous. ...
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Closed set through continuity

I have the measure space $(\mathbb{R}^2,\mathcal{B}(\mathbb{R^2}))$ and the set $A=\{x \in \mathbb{R}^2\mid w(x)\in[\theta, \eta], ||x||_2\in[r,R]\}$, where $0\le\theta\le\eta<2\pi, \text{and } ...
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Continuity of Derivatives

I am going over a statement in Rudin which says "Suppose $f$ is a real differentiable function on $[a,b]$ and suppose $f^{'}(a)<k<f^{'}(b)$. Then there is a point $x\in (a,b)$ such that ...
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25 views

Continuity of a function in the closed interval $[0,2]$.

Let, $g$ be a function defined on the interval $[0,2]$ and $x\le g(x) \le (x^{2}-x+1)$ for $0\le x \le 2$. Then, (1) $g$ must necessarily be a polynomial. (2) $g$ must be continuous at $x=1$. (3) ...
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56 views

Differentiability of a function on $\mathbb R$ such that $f(x+1)=f(x)$.

Let, $f:\mathbb R \to \mathbb R$ be a function such that $f(x+1)=f(x)$ for all $x\in \mathbb R$. Then which of the followings are correct? (a) $f$ is bounded. (b) $f$ is bounded if it is continuous. ...
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1answer
51 views

Let $f$ a continuous function on $\mathbb R$ such that $f(0)=f(2)$.

Let $f$ a continuous function on $\mathbb R$ such that $f(0)=f(2)$. Answer by true or false. There exists $\alpha\in[0,1]$ such that $f(\alpha)=f(\alpha+1)$. I think that it's wrong but I'm not able ...
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32 views

Show that f is uniformly continuous on [0, +∞)

So working on an exercise from my notes, I am given the conditions that $f$ is continuous on $[0, +∞)$ and uniformly continuous on $[a,+∞)$ for some $a > 0$. How do I show that $f$ is uniformly ...
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38 views

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$.

Using the Inverse Function Theorem prove that $(\sin^{-1}x)'$ = $\frac{1}{\sqrt{1-x^2}}$. Proof: Let $f(x) = \sin x$, for $x$ in $(-1,1)$. Then let $x_{0}$ be in (-1,1). Then $f'(x_{0})$ = ...
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Differntiable and continuous

Is it true that a function which is not continuous at a point will not be differentiable at that point? Graphically it seems so, but can we prove this formally? Also, if the above statement is ...
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Find the right derivative at a point using the definition

I have difficulties how to proceed this limit in order to find the value of this limit.
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Proof that Continuity Implies $\epsilon$-$\delta$ Criterion

(1) $\epsilon$-$\delta$ Criterion: For each $\epsilon > 0$, there is a $\delta > 0$ such that, for all $x$ in $\text{Dom}(f)$, $|x - c| \leq \delta \implies |f(x) - f(c)| \leq \epsilon$. ...
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Let $f(x)$ and $g(x)$ continuous functions at $x = 0$ such that $f(0) = 0 = g(0)$. Show that limit as $x$ approaches zero of $f(x)^{g(x)} = 1$

The test is very simple for the case $f(x) = x = g(x)$ since $$ \lim_{x\to 0}{x^x} = 1 $$ But in other cases? Note that they do not specify that the functions are differentiable and neither that they ...
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Derivative of a Decreasing Function

Show that if $C(K,T)$ is a differentiable function of $K$, then the derivative of $C(K,T)$ must lie between between minus one and zero. I have to use the following theorem: $C(K,T)$ is a decreasing ...
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61 views

If $f$ is a continuous odd function. Prove that if $f$ is differentiable at $0$, then there is a continuous even function $g$ such that $f(x) = xg(x)$

I'm working backwards to see if I can find the $g$, however, when I take the derivative of $xg(x)$ I have $f'(x) = g(x) + xg(x)'$ at $0$, then it will always ends up with $0$. Then I have no idea how ...
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Proving $f$ is uniform continuous

I am a little off my game today, so I can't immediately see a "way out" out of this question. If $f$ is continuous on $\Bbb R$ and $\lim_{x \to \pm \infty} f(x) = 0$, $f$ must be uniformly ...
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Is uniform continuity needed here…?

I found this problem, but I don't think uniform continuity is required. If $f$ is uniformly continuous on $(0,1)$, show $\lim_{x \to 0^{+}} f(x)$ exists. Doesn't this just fall from $f$ being ...
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365 views

What does continuity *in general* mean?

I am looking from : http://en.wikipedia.org/wiki/Lipschitz_continuity Continuously differentiable $\subseteq$ Lipschitz continuous $\subseteq$ α-Hölder continuous $\subseteq$ uniformly continuous ...
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Proof $f(x)\equiv 0$

Let $f\in C ((-\infty,+\infty)). $ If $ \forall a,b\in(-\infty,+\infty),\int_{a}^{b}f^{2}(x)dx \leq f(a)+f(b), $ then$f(x)\equiv 0$. I can prove $f(x)\geq 0,$ so I want use the reduction to ...
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Question about continuity function

Show that $f:A$ to $R$ is continuous on $A⊆R$ and if $n∈N$, then the function $f^n$ defined by $f^n (x)=(f(x))^n$, for $x∈A$, is continuous on A. Can anyone help me with this problem, thank you!
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Need help with a continuity proof

Let $f:R$ to $R$ be continuous on $R$, and let P:= {$x∈R: f(x)>0$}. If $c∈P$, show that there exists a neighbourhood $V_δ(c)⊆P$ Can I say: Let $ε>0$ be given, there exists $δ>0$ such that ...
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A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact

I'm trying to prove my sequence of functions $(f_n) = \frac{n}{n+1}\cos(x^2)$ on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of ...
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Continous function with one of ranges as an equation

I'm pretty new here and my formatting might have some errors, sorry I could get it only this far. f(x) = \begin{cases} \ ax+2b, & x<0 \\[3ex] \ x^2 + 3a -b, & x^2 + 3a -b \\ \ 3x-5, ...
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If $f$ is uniformly continuous on $(a, b)$, then $f$ is bounded on $(a, b)$.

So I know that since f is uniformly continuous on (a, b), then for every $\epsilon > 0$, there exists $\delta > 0$ such that for all x and y in (a, b), if |x - y| < delta, then |f(x) - f(y)| ...
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Continuity in Metric Spaces between two spaces under a function f

Let (X,d) and (Y,e) be metric spaces , and let f:X→Y be a function. Explain but do not prove if the statement is correct. If there exists r>0 so that $e((f(x1),f(x2))$$≤$ $r(d(x1,x2))$for every ...
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Extension of a function to a continuous function

The problem is the following: Extend the following function to a continuous function defined on all $\mathbb R^2$ $f(x,y)=xy/(x^2+y^2) $ I have never solved such a problem. Would be thankful if ...
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Prove that there exists continuous function of Sorgenfrey line to space $\mathbb{N}$…

Prove that there exists continuous function of Sorgenfrey line to space $\mathbb{N}$ with induced topology from euclidan topology on $\mathbb{R}$. I don't know if I understand the continuity ...
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Say $f : \mathbb{R} \to \mathbb{R}$ is continuous and $f(x) \to 0$ as $x \to \pm\infty$. Show $f$ is uniformly continuous.

I have a problem from Carother's Real Analysis, page 116. Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous and $f(x) \to 0$ as $x \to \pm\infty$. Prove that $f$ is uniformly continuous. ...
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Logic behind continuity definition.

I have a question regarding the definition of continuous functions : from wikipedia and my book : $f$ is said to be continuous at the point $c$ if the following holds: For any number ...
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Linear Continous Mappings

I am trying to prove the following Theorem: where $E$ and $F$ are normed vector spaces over the field $\mathbb{R}$ equiped with topologies introduced by means of their norms. I am not able to ...
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66 views

'Easy' question on continuity of integral

Here's a problem that I recently stumbled upon. It seems pretty easy, and quite intuitive yet every time I try to solve it, I run into some difficulties. Here it goes : Let $\phi \in ...
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28 views

Did I miss anything on this question about continuity on the value of $\alpha$

For what value of $\alpha$ is $f(x)$ differentiable at $x = 1$ For each of those values of $\alpha$, find $f^{\prime}(1)$ $$\mbox{Let } f(x) = \begin{cases} 2x-4\tan (x) & x \leq 0 \\ ...
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A sufficient condition to ensure a function to be linear

Suppose that $f$ is continuously differentiable on $\Bbb R$, and $$\lim_{x\to +\infty}f'(x)$$ exists and is finite. Furthermore, $$f(x+1)-f(x)=f'(x),\ \forall\ x\in\Bbb R.$$ Show that $f$ is linear, ...
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338 views

If $|f(x)-f(y)|\geq \frac12|x-y|$, must $f$ be bijective?

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $$|f(x)-f(y)|\geq \frac12|x-y|$$ for all$x,y\in \mathbb R$. Then is $f$ one-one and onto? Let $f(x)=f(y)$ i.e. ...
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right inverse and supplement of kernel in a banach

For $T \in L(E,F)$ continuous surjective linear operator between Banach spaces $E$ and $F$ we have that : $Ker(T) $ admits a closed complement $L$ in $E \implies T$ admits a continuous right ...
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$F \colon [a,b] \rightarrow \mathbb{R}$ continous with $F(a) = F(b)$ [closed]

Suppose $F \colon [a,b] \rightarrow \mathbb{R}$ is continous with $F(a) = F(b)$. How can I show that there are $c,d \in (a,b), c \neq d$ with $F(c) = F(d)$?
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24 views

Show that the image of a complete metric space under a continuous map is also complete given an additional condition.

This is a problem from revision material for a functional analysis class. Let $(X,d)$ and $(C,p)$ be two metric spaces and let $f:X\rightarrow C$ be a continuous function with $f(X)=C$. Assuming ...
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63 views

Continuity of the function defined by: $f(x)=e^x$ if $x$ is rational; $f(x)=e^{1-x}$ if $x$ is irrational

Let the function $f(x)$ be defined as $$f(x)= \begin{cases} e^x & x\text{ is rational} \\ e^{(1-x)} & x\text{ is irrational} \end{cases} $$ for $x$ in $(0,1)$. Then a. $f$ is ...
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1answer
58 views

Analysis-Baby Rudin's differentiability and continuity: theorem 5.2 and 5.6

I am very confused about differentiability and continuity. At the beginning of the differentiation chapter, we proved that differentiability contains continuity. (Theorem 5.2) But in example 5.6 and ...