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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92 views

Proving nonexistence of sequence of continuous functions convergent pointwise to Dirichlet function (definition only)

A fellow member of the community asked: "there isn't a sequence of continuous function on $[0,1]$ that converges pointwise to the function $f$ on $[0,1]$ defined by $f(x)=0$ if $x$ is rational and ...
6
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2answers
588 views

Is a continuous function between two uniformly continuous functions uniformly continuous?

I'm sorry for the long question in the title. Given three functions $\underline{f}(x), f(x), \overline{f}(x)$ that satisfy the following $\underline{f}(x)\leq f(x)\leq \overline{f}(x)$ for all ...
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1answer
63 views

Intermediate Value Theorem: Prove that there is $x$ such that $f(x+1)-f(x)=\frac{f(2)-f(1)}{2}$

We have continuous function $f:[0,2]\rightarrow \mathbb{R}$. Prove that there are $x_1,x_2$ such that $x_2-x_1=1$ and $f(x_2)-f(x_1)=\frac{f(2)-f(1)}{2}$ I was thinking about it a lot and can't find ...
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3answers
58 views

Does the identity map on a dense subset of a space extend uniquely?

Let $D$ be a dense subset of a (not necessarily Hausdorff) topological space $X$. Does the identity map on $D$ necessarily uniquely extend continuously to the identity on $X$? If not, what ...
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35 views

Prove that if $f$ is continuous on a compact set then it is uniformly continuous

Prove that if $f$ is continuous on a compact set then it is uniformly continuous. Proof: Let $f:A\rightarrow \mathbb{R}$ be a continuous function and let $A$ be a compact subset in a metric ...
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60 views

An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min ...
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0answers
31 views

Is an unbounded function bounded on a bounded non-compact interval?

I'm a little confused about functions in the set of bounded continuous functions. For example, if we take the interval (0,1] and the function $f(x) = $\begin{cases} 0 & x \in ...
2
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2answers
63 views

Does Darboux theorem imply that $f'$ cannot have jump discontinuity?

Does the Darboux theorem for derivatives imply that a derivative on a interval $I$ cannot have jump discontinuity? Darboux theorem states that the derivative function follow the intermediate value ...
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2answers
61 views

Lipschitz function and continuously differentiable function

If function $f$ is continuously differentiable at some point, say $x=0$, and is Lipschitz in some neighborhood of $x=0$, is that true there is an open neighborhood of $x=0$ in which $f$ is ...
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0answers
75 views

Prove that $\lim_\limits{x\to 2}f(x)=3.$ [duplicate]

Let: $$f(x) = \begin{cases} 5-x, & \text{if $x$ is irrational.} \\[2ex] 1+x, & \text{if $x$ is rational.} \end{cases}$$ Prove that $\lim_\limits{x\to 2}f(x)=3.$ Prove that ...
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2answers
31 views

Absolute continuity, shifted set

Let $\mu$ be a finite signed measure on the Borel sets of $\mathbb{R}$, and suppose that $\mu \ll m$ where $m$ is Lebesgue measure. Prove that the function $t\mapsto \mu\{t+x:x\in A\}$ is ...
1
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1answer
29 views

Why is a von Neumann algebra is closed with respect to weak * topology?

I was trying to prove that the identity map between a von Neumann algebra $(A,\mbox{ultra weak topology})$ with respect to ultra weak topology and the von Neumann algebra $A$ with respect to weak* ...
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0answers
37 views

Meaning of a theorem regarding the limits of derivatives?

I'm confused about this theorem which is sometimes associated to the Darboux theorem for real functions. Let $f: dom(f)\subseteq\mathbb{R} \rightarrow \mathbb{R}$ be a function continuous in a point ...
0
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1answer
75 views

Function continuous only in a point

Is true that the function $f:\mathbb{R}\rightarrow \mathbb{R}$ $f(t):=\begin{cases}t \: , \: t\in \mathbb{R}-\mathbb{Q} \\ 0\: , t\in \mathbb{Q} \end{cases}$ is continuous only in the point $0$ ? ...
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1answer
46 views

Identifying a property of contiuous of functions and its proof

I am reading a proof of the EVT (http://math.umn.edu/~kling202/hamline/calculus/Chapter4/EVTProof.pdf) and I came across this property of continuous functions (i.e., taking the limit outside of the ...
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2answers
22 views

Range of continuous transformation on closes set

let $f$ be a continuous transformation and $F$ closed set. Prove that the range $f(F)$ does not have to be closed.
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1answer
20 views

Localization of ring of continuous functions at an element

Let $X$ be a topological space. Let $D_f= \left\{ x\in X:f(x)\neq 0 \right\}$. Let $\mathcal O_X$ be the sheaf of continuous functions on $X$. Is it true that $\mathcal O_X(D_f)= \left\{ \frac ...
0
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1answer
29 views

Prove that there is no continuous injective function from closed rectangle in $\mathbb{R}^2$ to $\mathbb{R}$.

Prove that there is no continuous function from a closed rectangle in $\mathbb{R}^2$ to $\mathbb{R}$ that is injective. How can I start?
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3answers
301 views

Give another proof of intermediate value theorem

Give another proof of intermediate value theorem by completing the following argument: If $f$ is a continuous real-valued function on the closed interval $[a,b]$ in $\mathbb{R}$ and ...
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0answers
14 views

tweak on the implicit function theorem

I am studying solutions to an equation of the form $y(x,v)=0$ where $x,v$ are reals. $y$ is continuous in both its arguments. The implicit function theorem says that if a solution exists to the above ...
0
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1answer
26 views

Does $f''(0^+)=f''(0^-)$?

Consider the function, which is the join of two semicircles $$ f(x) = \left\{ \begin{array}{cc} \sqrt{1 - x^2} & x > 0 \\ 1 & x = 0 \\ \sqrt{ 2 - (x-1)^2} & x < 0\end{array} ...
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0answers
34 views

If number of points of discontinuity of the function $f(x)=\lfloor{2+10 \sin x\rfloor}$,in $[0,\frac{\pi}{2}]$ is same as

If number of points of discontinuity of the function $f(x)=\lfloor{2+10 \sin x\rfloor}$,in $[0,\frac{\pi}{2}]$ is same as number of points of non-differentiability of the function ...
3
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1answer
110 views

Population dynamics

I don't understand why we make the three assumptions underlined above.
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2answers
118 views

Proving using the mean value theorem

Definition: A function is said to be periodic with period $p>0$ if for every $x\in\mathbb{R}:f(x+p)=f(x)$. Prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous and $p$-periodic, then it has a ...
3
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1answer
43 views

Why is a characteristic function continuous at $0$?

My lecture notes say: $t \mapsto \exp(-t^2/2)$ is a characteristic function (of $\mathcal{N}(0,1)$), so it is clear that it is continuous at $0$. So why does "being a characteristic function" ...
0
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1answer
35 views

continuous functions, product topology

I have to prove the following statement: $(X,\mathcal{T}_X), (Y,\mathcal{T}_Y), (Z,\mathcal{T}_Z)$ topological spaces. $h=(f,g):Z\rightarrow X\times Y, h(z)=(f(z),g(z))$ is continuous if and only if ...
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3answers
38 views

Classify the type of discontinuity at $x_0 = 0$

for (a) I think it is essential because the right side goes to infinity. for (b) I think it is removable because the function is not defined in $0,$ same goes for (c) I am really not sure about ...
3
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2answers
42 views

Have anyone ever thought of continuous analog Turing machine?

Have anyone ever thought of continuous analog Turing machine? The machine adopts continuous (from R) the input data from the tape, It moves to a different state depending on the value on the tape. On ...
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1answer
17 views

Inserting a Lipschitz function between two continuous functions

I am reading a proof that uses the following fact which I cannot show: Let $b_1, b_2 : \mathbb{R} \rightarrow \mathbb{R}$ be two continuous functions such that $b_1 (x) < b_2 (x),$ for all $x ...
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1answer
25 views

Implications of continuous differentiability at a point

Consider a function $f:\mathbb{R}^l\rightarrow \mathbb{R}$ continuously differentiable at $x_0$. This implies that (1) the function is differentiable on a neighbourhood of $x_0$ which means that the ...
2
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1answer
16 views

Can a point and a compact set in a Tychonoff space be separated by a continuous function into an arbitrary finite dimension Lie group?

Given a topological space $X$ which is Tychonoff (i.e., completely regular and Hausdorff), we know that given a compact set $K\subseteq X$ and a point $p \in X$ with $p\not\in K$, we can construct a ...
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2answers
93 views

Prove $f(x)=2^x$ is continuous

I have show that $f(x)=2^x$ is continous by using the Weierstrass definition (epsilon-delta). I set apart two cases. The first one $x>x_0$ was good. The second one is the problem right now. ...
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2answers
58 views

Does it follow that $f:\Bbb R^n\to \Bbb R^m$ is bounded in every bounded interval $I\subseteq \Bbb R^n$ if $f$ is continuous in $\Bbb R^n$?

I was trying to solve a bigger exercise and I thought of the following Conjecture Let $f:\Bbb R^n\to \Bbb R^m$ be continuous in $\Bbb R^n$. Then $f$ is bounded in every bounded interval ...
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1answer
50 views

Is a “differentiable” equation really guaranteed to be continuous?

There is this theorem I just learned that states a differentiable (everywhere) function is also continuous everywhere. Here is a popular proof for it; ...
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31 views

Image of a continuous function from the reals to the reals

I know the image of a connected set in $\mathbb{R}$ under a continuous function to $\mathbb{R}$ is also a connected set- I.e intervals map to intervals. But what about the image of a disconnected set? ...
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1answer
63 views

Exchanging limits and Riemann Integral

Consider a function $f:\Theta \subseteq \mathbb{R}\rightarrow \mathbb{R}$ continuous at $\theta=\theta_0$ such that $f(\theta)\geq 0$ $\forall \theta \in \Theta$. Consider a sequence of real numbers ...
0
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1answer
21 views

Integral of a continuous function of l+1 variables

Consider a function $f(\theta, x): \Theta \times \mathcal{X} \rightarrow \mathbb{R}$ where $\mathcal{X} \subseteq \mathbb{R}$ and $\Theta \subseteq \mathbb{R}^l$. Suppose the map $\theta \rightarrow ...
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2answers
68 views

How to prove this simple limit?

Let $f:[0,+\infty]\to R^1$ to be a function which satisfies the following properties. $$ $$ 1.$f$ is uniformly continuous on $[0,+\infty)$ $$ $$ 2.For any $x_0\in[0,1]$,we always have ...
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3answers
61 views

If $f(x.y)=f(x).f(y)$ for all $x,y$ and $f(x)$ is continuous at $x=1$,then show that $f(x)$ is continuous for all x except at $x=0$.Given $f(1)\neq 0$

If $f(x.y)=f(x).f(y)$ for all $x,y$ and $f(x)$ is continuous at $x=1$,then show that $f(x)$ is continuous for all x except at $x=0$.Given $f(1)\neq 0$ In the functional equation ...
0
votes
1answer
29 views

Struggling with the concept of continuity in high school calculus

I am a high school student and I am slightly confused regarding certain aspects of continuity in my calculus class. Rational functions are often given as examples of functions which possess so-called ...
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3answers
54 views

Continuity of improper integral with a continuous integrand.

I am a newbie in analysis and am trying to wrap my head around some continuity/compactness/finiteness concepts. Let $f(x,y):\mathbb{R}^2\mapsto\mathbb{R}$ be a continuous function in both $x$ and $y$ ...
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2answers
87 views

Extend a continuous function on $(0,\infty)$ to a function on $[0,\infty)$

Which of the following given functions $f:(0,\infty)\rightarrow \mathbb R$ can be extended to become a continuous function on $[0,\infty)$? $\sin{1\over x}$ ${1-\cos x}\over x^2$ $\cos {1\over x}$ ...
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0answers
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Jointly continuous on product topology versus standard topology

Sorry this is a laymen question. I commonly see references to a function of two variables as being 'jointly continuous' especially in proofs using homotopies. I sometimes get confused as to which type ...
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64 views

Prove that $f_n(x)=\frac{x}{n}$, $n=1,2,\ldots$ does not converge uniformly on $\mathbb{R}$

Prove that $f_n(x)=\frac{x}{n}$, $n=1,2,\ldots$ does not converge uniformly on $\mathbb{R}$. It's clear this function converges pointwise to $0$ function. We have to show that there is ...
5
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1answer
83 views

Why do we take the closure of the support?

In topology and analysis we define the support of a continuous real function $f:X\rightarrow \mathbb R$ to be $ \left\{ x\in X:f(x)\neq 0\right\}$. This is the complement of the fiber $f^{-1} \left\{0 ...
3
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2answers
156 views

Continuity function

Let f be a continuous real function such that $f(11)=10$ and for all $x$, $f(x)f(f(x))=1$, then what is the value of $f(9)$. I got $$f(10)=\frac{1}{10}$$
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2answers
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Whether a continuous function has fixed point or not when the domain and range are not $[0,1]$

Which of the following is false $?$ $A.$ Any continuous function from $[0,1]$ to $[0,1]$ has a fixed point. $B.$ Any homeomorphism from $[0,1)$ to $[0,1)$ has a fixed point. $C.$ Any bounded ...
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1answer
17 views

Which Properties of a Natural Cubic Spline does the following function possess and not possess

I need to determine which of the properties of a natural cubic spline the following function possesses or does not possess: $$f(x) = \begin{cases} (x+1)+(x+1)^{3}, & x \in [-1,0] \\ ...
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1answer
47 views

$f:\mathbb R\rightarrow [0,\infty )$ is continuous such that $g(x)={(f(x))}^2$ is uniformly continuous .

$f:\mathbb R\rightarrow [0,\infty )$ is continuous such that $g(x)={(f(x))}^2$ is uniformly continuous . Then which of the following is always true $?$ $A.$ $f$ is bounded. $B.$ $f$ may not be ...
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2answers
25 views

The points where function is discontinuous,are those points counted/considered in the domain of the function.

The points where function is discontinuous,are those points counted/considered in the domain of the function. $(1)[x]$,greatest integer function is discontinuous at all integer points but integers are ...