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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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)} ...
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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 ...
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22 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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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 ...
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51 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 ...
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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 ?
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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 ...
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35 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 ...
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23 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
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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 $\displaystyle \lim_{x \rightarrow t + } f(t) , \lim_{x \rightarrow t - } f(t)$ ...
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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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32 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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7 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 $u, u_t, u_{x_i}$ ...
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26 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 ...
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18 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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26 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 ...
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3answers
75 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 ...
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267 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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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 ...
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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 ...
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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 ...
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25 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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11 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 ...
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4answers
61 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 ...
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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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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 ...
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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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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 ...
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1answer
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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
38 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 ...
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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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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 ...
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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 ...
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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 ...
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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' ...
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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 ...
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35 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 ...
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1answer
30 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 ...
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21 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 $a_n$?
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11 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 ...
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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 ...
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39 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 $$1-\cos(x)=2\sin^2(x/2)$$ ...
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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?
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1answer
31 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 $0$. ...
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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 ...
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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. ...
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28 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 } ...