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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132 views

Is a differentiable function always integrable?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
0
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0answers
32 views

Example of a continuous function such that f(X)=Y.

Can anyone give me an example of a continuous function $f$ from $X=[0,1] \cup [2,3]$ to $Y = \{ 0,1 \}$ such that $f(X)=Y$? Or alternatively can you explain why such a function does not exist? ...
2
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1answer
17 views

Example of continuous functions $f\colon S \to T$ such that $f(S)=T$.

I would like to find an example of a continuous function from $S=(0,1)$ to $T=(0,1)\cup (1,2)$ such that $f(S)=(T)$. At the moment the only thing I can think might work would be to check whether ...
2
votes
3answers
45 views

If $f: \mathbb{R} \to \mathbb{R}$ is continuous then $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$

Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$. At first I thought this was quite ...
2
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1answer
15 views

some statements based on continuity

$f:\mathbb{R}\to\mathbb{R}$ is continuous and injective, then it is strictly monotone. True If $f\in C[0,2]$ with $f(0)=f(2)$, then $\exists x_1,x_2\in [0,2]\ni x_1-x_2=1$ and $f(x_1)=f(x_2)$ ...
0
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1answer
53 views

Is the definition of continuity in analysis a particular case of topological continuity?

Take a constant function and remove an open interval from it: $$f(x)= 1, \text{if $x\in(-\infty,0]\cup[1,\infty)$ }$$ This function shouldn't be continuous because at $0$ no right limit of the ...
3
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1answer
25 views

Directly proving continuous differentiability

Let us say that we want to prove that a function $f: I \to \mathbb{R}$ defined on an open interval $I$ is continuously differentiable on $I$. One way to do this is to establish that $f'(x)$ exists at ...
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2answers
36 views

Discontinuous everywhere but range is an interval

Does there exist a function which is discontinuous everywhere but range set is an interval.
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0answers
16 views

poisson process and continuity [on hold]

This question is about poisson process and its continuity. N = (Nt)t≥0 represents a Poisson process of positive rate λ How to show the limit of P(|Nt −Ns|>ε) tends to 0 when s tends to t?
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1answer
27 views

Checking continuity looking whether image set is interval or not

Let $A(\neq \phi)\subseteq\mathbb{R}$. Suppose $f : A \to \mathbb{R}$ is a monotone function such that the image $f (A)$ is an interval. Then prove that $f$ is a continuous function. And if $f(A)$ is ...
2
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1answer
18 views

Is the function continuous that maps a vector to the coefficients of its expansion in a basis?

Let $V$ be a finite-dimensional vector space with basis $e_1, e_2, \ldots, e_n$. Consider the function $f:V\rightarrow\mathbb{R}^n$ defined such that, for each $v\in V$, $f(v)=(a_1,a_2,\ldots,a_n)$ ...
4
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3answers
68 views

Continuous functions satisfying $f(x)+f(2x)=0$?

I have to find all the continuous functions from $\mathbb{R}$ to $\mathbb{R}$ such that for all real $x$, $$f(x)+f(2x)=0$$ I have shown that $f(2x)=-f(x)=f(x/2)=-f(x/4)=\cdots$ etc. and I have also ...
3
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1answer
75 views

If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$

Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints. I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} ...
0
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0answers
39 views

problem with deriving continuity equation

I am studying Aerodynamics, to be more precise, the fundamentals of Aerodynamics. The first law is the continuity equation, for which it is explained in the book that I am using. However, I wished to ...
1
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1answer
25 views

$f:X\to Y \text{ is continuous} \iff f^{-1}(A^*) \subseteq (f^{-1}(A))^*$

Really struggling with exercise 9.10 from Sutherland's "Introduction to Metric and Topological Spaces". Any help would be greatly appreciated. Let $(X,t), (Y,t)$ be topological spaces, and $f: X \to ...
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0answers
18 views

Real analysis/cont-UC [on hold]

I have exam tomorrow and i need a help If $df/dx$ is bounded on any interval $E$ then $f$ is Uniformley continuous ? What about if $E$ is compact?Is it true or false ?Justify or give an example ...
0
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1answer
53 views

Is there a topological proof that additon and multiplication are continous functions from $\mathbb R \times \mathbb R $ into $\mathbb R $?

Is there a topological proof that additon and multiplication are continous functions from $\mathbb R \times \mathbb R $ into $\mathbb R $? That is, can we prove continuity using the topological ...
2
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0answers
37 views

Looking for differentiable function $f:\mathbb R \to \mathbb R$ whose derivative is nowhere continuous [duplicate]

Does there exist a differentiable function $f: \mathbb R \to \mathbb R$ such that its derivative $f'$ is nowhere continuous ?
0
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2answers
54 views

Countable subset and monotonic function

let E be subset of R which has no isloated points(or C does not have any isolated point of E) and C be countable subset of R does there exist a monotonic function on E which is continuous only at ...
3
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0answers
37 views

Proving continuity on Sobolev space with weak topology

Hi I am interested in proving that an operator $$\eta : W^{1,p}(\Omega) \times L^{p}(\Omega) \rightarrow L^{p'}(\Omega)$$ is (weak $\times$ norm, norm) continuous. I want to know if it is viable to ...
0
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0answers
26 views

Complete subspace of continuous function from compact subset [on hold]

Assume $K\in \mathbb{R}$ compact. How to prove that $C^0(K,\mathbb{R})$ is complete. Where $C^0(\mathbb{R},\mathbb{R})$ is the space of continuous f from $\mathbb{R}$.
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2answers
42 views

Counterexample for “if every continuous function on $E$ is bounded, then $E$ is compact” [on hold]

Give me counter example for this false statement: "Every continuous function on the set $E$ is bounded this implies $E$ is compact".
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1answer
25 views

Finite number of jump discontinuities

Let $f : (a,b) \rightarrow \mathbb{R}$ be a monotonic function. $t \in (a,b)$ is called a jump discontinuity of $f$ if $\displaystyle \lim_{x \rightarrow t + } f(t) , \lim_{x \rightarrow t - } f(t)$ ...
0
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1answer
22 views

Conditions for open interval continuity

Please can someone help in giving me the condition that would make a continuous function on an open interval be uniformly continuous in that same interval. Thanks.
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2answers
35 views

If a continuous function has one relative extremum, then it is actually an absolute extremum

The following statement makes sense intuitively, but is there a way to prove it mathematically? (This is something we make use of in applied optimization in calculus.) If $f$ is continuous on an ...
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0answers
8 views

Space $C^2(\overline{U})$ for open set $U$

Let $U$ be a bounded open domain in $\mathbb{R}^n$. Does the space $C^2(\overline{U})$ (the bar over $U$ means closure) mean the set of twice-differentiable functions $u$ such that $u, u_t, u_{x_i}$ ...
1
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0answers
32 views

Multivariate Weierstrass theorem?

The Weierstrass theorem states that for any continuous function $f$ of one variable there is a sequence of polynomials that uniformly converge to $f$. To my surprise, I couldn't find any reference to ...
0
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0answers
20 views

A random variable with distribution continuous on a parameter: Is a continuous function of such random variable continuous in the parameter?

Let $(X_n(\lambda))_{n\in\mathbb{N}}$ be a sequence of i.i.d. real continuous random variables (with density function) and assume that $P(X_n(\lambda)\le x)$ is continuous in $\lambda$. Consider the ...
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1answer
44 views

Thomae's function, doubt in continuous proof in the irrationals.

I was studying about this proof and i almost understand all of it, i just have one doubt there, the proof i found is the following; Let f be defined by; $$ \begin{align} f(x) = \begin{cases} 0 & ...
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2answers
28 views

continuity of functions on intervals

Suppose that $f : (a,b) \to \mathbb R$ is continuous. Then, there is a continuous $g : [a,b] \to \mathbb R$ such that $g(x) = f(x)$ for all $x \in (a,b)$. That is, a function defined and continuous on ...
0
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3answers
76 views

$f: \mathbb{R} \to \mathbb{R} $ by $f(x) =\frac 1{1+x^2}$ is uniformly continuous on $\mathbb{R} $

The definition of uniform continuity states that a function is uniformly continuous if, given any challenge $\epsilon > 0$ that there exists a response $\delta > 0$ for every value $x_1,x_2 \in ...
3
votes
2answers
268 views

Mapping 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} $ ...
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2answers
23 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
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3answers
36 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
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1answer
47 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
26 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) − ...
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0answers
12 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
62 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
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1answer
34 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
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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
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1answer
40 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 ...
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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
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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 ...
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2answers
39 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
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1answer
41 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.
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2answers
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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 ...
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1answer
28 views

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

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
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1answer
40 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
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0answers
25 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
33 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' ...