Tagged Questions

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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3
votes
2answers
210 views

A continuous function from 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} $ ...
1
vote
2answers
21 views

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 ...
0
votes
3answers
35 views

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 ...
0
votes
1answer
46 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 ...
2
votes
2answers
22 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) − ...
0
votes
0answers
10 views

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 ...
1
vote
4answers
60 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 ...
0
votes
1answer
31 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
0
votes
1answer
10 views

Multiple choice question on a fixed point of a continuous function

$f$ is a continuous mapping from $[0,1]$ to itself which is continuously differentiable in $(0,1)$ and such that $|f^{'}(x)|\leq 1/2 \forall x\in (0,1)$.Then there exists a unique $x\in [0,1]$ such ...
1
vote
1answer
32 views

Proving that the function f is of class C^1,

Suppose $f:R->R$ is continuous, and that it has a continuous right derivative, i.e. the right-sided limit $$lim(\delta->0^+) (f(x+\delta)-f(x))/\delta$$ exists for all x $\in$ R and defines a ...
0
votes
2answers
16 views

Does the function of a bounded sequence have a convergent subsequence?

Let {$x_n$} be a sequence in (s,t), and suppose f is continuous on [s,t]. Then does {f$(x_n)$} have a convergent subsequence? I know if {$x_n$} converges to some $x_0$ then {f$(x_n)$} converges to ...
0
votes
1answer
13 views

Multiple choice question on continuous function on a unit ball

Pick out true: Let $B$ be the closed unit ball and $D$ be the open unit ball. a.Given a continuous function $g:B\rightarrow \mathbb R$ there always exists a continuous function $f:\mathbb ...
1
vote
2answers
37 views

Does the presence of irrational numbers pose any problems for the concepts of limits and continuity?

Could someone discuss in an intuitive (not too formal) way whether irrational numbers like $\pi$ would pose any problems to the ideas of limits and continuity? I'm not sure if they do, or not, but it ...
0
votes
1answer
32 views

$\lim_{|x|\to\infty}f(x)=0$ implies $f$ attains its maximum value

If we suppose that $f$ is a positive continuous function on $\mathbb{R}^n$ such that $\displaystyle\lim_{|x|\to\infty}f(x)=0$. I want to show that $f$ attains its maximum value.
0
votes
2answers
24 views

Use induction to show that maximum of $k$ real-valued continuous functions is continuous.

For this question I must use induction to show that if $f_i$, $i=1, \cdots, k$ are continuous real-valued functions on $S$, then $$h(x)=\max_{i=1, \cdots, k} f_i(x)$$ is continuous. So I am not ...
-1
votes
1answer
24 views

Show that m(x,y)=max{x,y} is continuous on R^2 [on hold]

I am required to show that m(x,y)=max{x,y} is continuous on R^2 and then part b) Hence show that if f and g are continous real-valued functions on a set S element R^n, then h(x)=max{f(x),g(x)} is ...
2
votes
1answer
39 views

Vector-valued function, proving whether it's continuous, based on its action on any line in R^2:

Suppose $f: R^2 -> R^2$ is a function whose restriction to any line L in $R^2$ is continuous. Prove or find a counterexample: f must be continuous. For starters, I drew an arbitrary point on the ...
2
votes
0answers
23 views

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 ...
0
votes
1answer
31 views

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' ...
0
votes
1answer
22 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 ...
0
votes
2answers
33 views

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 ...
1
vote
1answer
28 views

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 ...
0
votes
2answers
20 views

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 $a_n$?
0
votes
0answers
10 views

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 ...
0
votes
0answers
22 views

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 ...
1
vote
2answers
38 views

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 $$1-\cos(x)=2\sin^2(x/2)$$ ...
1
vote
1answer
31 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?
1
vote
1answer
29 views

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 $0$. ...
1
vote
1answer
30 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 ...
2
votes
1answer
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. ...
1
vote
0answers
25 views

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 } ...
0
votes
0answers
20 views

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 ...
0
votes
1answer
23 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) ...
0
votes
3answers
34 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. ...
1
vote
1answer
50 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 ...
2
votes
1answer
27 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 ...
5
votes
1answer
35 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 $f'(x_{0})$ = ...
1
vote
3answers
32 views

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 ...
-1
votes
0answers
26 views

proof: $f ( x,y,z ) = x^2 + y^ 2 + z ^2$ is continuous function [closed]

proof: $f ( x,y,z ) = x^2 + y^ 2 + z ^2$ is a continuous function as the title say, thanks
0
votes
0answers
19 views

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.
-3
votes
0answers
48 views

Can anybody help me solve this exercise? [closed]

I want someone to solve this!!
1
vote
0answers
19 views

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 $\text{Dom}(f)$, $|x - c| \leq \delta \implies |f(x) - f(c)| \leq \epsilon$. ...
1
vote
3answers
39 views

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 ...
1
vote
0answers
25 views

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 ...
2
votes
4answers
48 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 ...
1
vote
2answers
29 views

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 ...
0
votes
2answers
25 views

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 ...
2
votes
4answers
355 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 ...
3
votes
1answer
56 views

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 ...
3
votes
2answers
36 views

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!