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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Continuous indicator-like functions

Let $\Omega$ be a compact subset of $\mathbb{R}^n$. Let $g:x\in\mathbb{R}^n\to\mathbb{R}$ be a continuously differentiable function such that $$ \begin{cases} g(x)>0 & x\in\text{int}\Omega,\\ ...
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Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable

Problem : Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable? For the sake of generality, let's assume that it is unknown to us that $|x|$ is not differentiable at $x = 0$ ...
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Proving a function $f$ is not differentiable at an unkown point $a$

Let's say I have an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and I want to prove that it is not differentiable at some unknown point $a$. Emphasis must be placed on the unknown part as that ...
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1answer
42 views

Question on limit of a function of a sequence

Let $f$ be a continuous real valued function on $[0,+\infty)$. Let $A$ be the set of real numbers $a$ that can be expressed as $$a = \lim_{n \to \infty}f(x_n)$$ for some sequence $(x_n)$ in $[0,+\...
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89 views

Alternate definition of differentability at a point

Usually in most introductory Calculus courses, a definition of differentiability at a point $a$ is defined, as follows : A function $f$ is differentiable at $a$ if $f'(a)$ exists As a corollary ...
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71 views

Can we define a metric on $Y$ such that all continuous mappings $f:X\rightarrow Y$ are constant?

Given that $Y$ contains more than one element and let $X$ be the real line equipped with the standard metric. Then can we define a metric $\sigma$ on $Y$ such that every continuous mapping $f:X\...
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49 views

Prove continuity of piecewise function using epsilon-delta

Suppose we have a function $\phi$ so that $$\phi (x)=\cases{f(x) & \text{ if } x\le 0\\ g(x)& \text{ if } x>0.}$$ where $f$ is continuous on $(-\infty,0]$ and $g$ is continuous on $(...
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1answer
11 views

$f( \phi^{-1}(x_0 +h)) = f(\phi^{-1}(x_0+h))+h \alpha+ O(h^2)$ - value of $\alpha$

Consider a function (continuous) $f : M \to \mathbb{R}$ with $M$ a $1$-dimensional manifold, and suppose some (smooth) chart $\phi : T \to \mathbb{R}$ having an (smooth) inverse on some open $(x_0- \...
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53 views

Periodic function with the capacity of being $g'' = \lambda g$

Related to the question : Eigenvalues of the circle over the Laplacian operator, what kind of periodic chart $c:(-\pi,\pi)\rightarrow S^1$ has the property that for a continuous function $f$, $g :=f \...
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3answers
41 views

Proof continuity of a function with epsilon-delta

I quickly need help with a problem that seems to be fairly easy but I can't really do the final step: Proof that the function $\frac{x-1}{x²+1}$ is continuus in $x = -1$ using the epsilon-delta-...
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Does a analytic joint distribution necessarily have continuous marginals?

Although the question actually popped up in a course about evolution equations, it seemed most natural to ask this in the context of joint distributions. Namely: Suppose $X$ and $Y$ are two ...
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1answer
45 views

Topology - $f$ is continuous iff $f$ is constant

Let $X_1$ be with the trivial topology, $X_2$ be Hausdorff, $f:X_1 \rightarrow X_2$ a function. Then $f$ is continuous $\iff f$ is constant. I'm not sure that my proof is correct so would appreciate a ...
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1answer
19 views

continuity of function series

So here it is: $$\sum_{n=1}^\infty \frac{\sin(\frac{1}{nx^2})}{1+(x-1)\ln^4(xn)}$$ $$x \in (1,\infty)$$ My task is to prove its continuity if possible. My lead was to try proving it through ...
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59 views

Claim about holomorphic extension

Prove or disprove the following claim. "For all continuous $f : S(0, 1) \to R$, there is a holomorphic $g : B(0, 1) \to C$ which extends to a continuous ${h : \overline {B(0, 1)}} \to C$ such ...
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Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
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The product topology on $X_1 \times X_2$ - coarsest due to some continuity

Let $X_1, X_2$ be topological spaces and $X_1 \times X_2$ with the product topology. We define the projection map $\Pi_i : X_1 \times X_2 \rightarrow X_i, \Pi_i(x_1,x_2) = x_i$. Consider the following ...
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1answer
45 views

A bounded non Riemann integrable real function with set of discontinuity of empty interior

Is possible to construct a bounded non Riemann integrable real function such that the set of discontinuity points has empty interior? I know that if the set of discontinuity points is a null set then ...
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1answer
38 views

Find continuous function $f$ with $f(\mathbb{Q}) = 0$ and $f(\mathbb{Q}+ \sqrt{2}) = 1$

Urysohn's Lemma approximates indicator functions with continuous functions. Let $X$ be a normal topological space. For every disjoint pair of closed sets $A,B$ there is a continuous function $f: ...
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1answer
49 views

$f(x)=\tan x$ for rationals, $f(x)=x^2+1$ for irrationals At exactly how many points will $f(x)$ be continuous within $[0, 6 \pi]$

$f(x)=\tan x$ for rationals, $f(x)=x^2+1$ for irrationals At exactly how many points will $f(x)$ be continuous within $[0, 6 \pi]$ I got the answer as $6$,am I correct?
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1answer
38 views

a function that continuous at every irrational but discontinuous at rational

Does exist a function $f$ that discontinuous at rational and continuous at every irrational but the restriction $f$ to the set of all irrational numbers is not constant and $f(q_n)$ is convergent ...
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1answer
26 views

Continuity of a function with a product as domain

Let $f\colon \mathbb{R}\times\mathbb{R}\to \mathbb{R}$ be a function such that the following holds: For every $x,y\in\mathbb{R}$, the functions $f(x,.)\colon\mathbb{R}\to \mathbb{R},$ $f(.,y)\colon\...
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8 views

Limit and Function defined at a Point of Discontinutiy.

Find the value of a that makes the following function continuous on $(-\infty, \infty)$. $f(x)= \frac{4x^3+13x^2+13x+30}{x+3}$ if $x\lt-3$, $5x^2+3x+a$ if $x \ge -3$}?
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1answer
23 views

Continuity vs differentiability versus directional derivatives

I'm having trouble with understanding the different concepts of continuity, differentiability and the existence of directional derivatives. I am given a function $f:\mathbb{R}^2\rightarrow\mathbb{R}, ...
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73 views

$X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?

Let $X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then is it true that the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?
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21 views

Prove: If f and g are two uniformly continuous functions in I, then $\alpha f+\beta g$ is also uniformly continuous in I

Prove: If f and g are two uniformly continuous functions in I, then $\alpha f+\beta g$ is also uniformly continuous in I Where $\alpha , \beta \in R$ and I is a section that can be closed or not. ...
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1answer
10 views

Name for function that is Lipschitz continuous over partitioning of input space

Let $f: X \to \mathbb R$ and $(X,d)$ be a metric space. Let $P=\{P_1,P_2,\dotsc\}$ be a countable partitioning of $X$. I would like to assume that $f$ is Lipschitz continuous on $(P_i,d)$ for all $P_i ...
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1answer
61 views

Prove that $f(x)=\frac{1}{x}$ is not uniformly continuous on (0,1)

So I'm having difficulties understand and utilizing the definition of uniform continuity: $\forall \epsilon \gt 0,$ $\exists \delta>0 $ such that $$ |x_1-x_2|\lt \delta \Rightarrow |f(x_1)-f(...
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2answers
38 views

Does Riemann integral of everywhere continuous and nowhere differentiable functions (with chosen values at the boundary points) can attain any value?

Suppose that we choose some interval and fix it, for example let us choose interval $[0,1]$. If $f$ is some everywhere continuous and nowhere differentiable function defined on $[0,1]$, then, because ...
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21 views

Is the Restriction of a Continuous Map again a Continuous Function?

Is true that if $g:Y\to Y$ is continuous then a mapping $f:X\to Y$ with $X \subset Y$ is continuous? I think it's true. Since for every open set $U$ in $Y$, we have that $g^{-1}(U)$ is open in $Y$. ...
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If $S$ is not compact, there is a continuous function unbounded on $S$

problem This was given to me as a homework problem to prove: If $S \subseteq \mathbb{R}$ is not compact, then there exists a continuous function $f : S \rightarrow \mathbb{R}$ that is unbounded ...
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2answers
34 views

Constructing a sequence of functions, not Cauchy

I'm working in the set $B = \{ f \in C[0,1] : \int_0^1 f(x)dx \leq 1\}$. I'm constructing an argument to show that there exists at least one sequence that has a subsequences satisfying the property ...
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1answer
43 views

“Continuous maps are those maps that do not tear space apart”

In a tutorial I wanted to give a quick explanation of the property of continuity. One of the common intuitions for continuity is that it preserves connection: Continuous maps do not map connected ...
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Linear programming is continuous

Consider an arbitrary linear program: $$\max \vec c \cdot \vec x$$ subject to: $$\textbf{A}\cdot \vec x = 0, \quad \vec a \le \vec x \le \vec b$$ Assume that this program is feasible and bounded. ...
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Using $\epsilon-\delta$ proof to prove continuity

Use an $\epsilon-\delta$ proof to show that $f : R \setminus \left \{ \frac{-3}{2} \right \} \rightarrow R$ , $$f(x) = \frac{3x^2-2x-5}{2x+3}$$ is continuous at $x = -1$ Hello there. Can anyone ...
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1answer
51 views

Rigor in proving continuity of $f$ over a closed interval $I$

Given a function $f$ on a closed interval $I \subset \mathbb{R}$, where $I = [a,b]$, to prove continuity of $f$ over the interval $I$, what is generally done is the following. 1. We prove that $f$ is ...
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2answers
42 views

Is each function $A^{\mathbb{Z}}\to A^{\mathbb{Z}}$ continuous?

Let $A$ be some finite alphabet. Let $A$ be equipped with the discrete topology and $A^{\mathbb{Z}}$ equipped with the associated product topology. Am I right that each function $f\colon A^{\mathbb{...
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3answers
31 views

Continuous for each variables does not implies continuous

Prove or disprove the following statement: Statement. Continuous for each variables, when other variables are fixed, implies continuous? More clearly, prove or disprove the following problem: Let $\...
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1answer
78 views

A smooth nowhere analytic function such that all derivatives are monotone

Related questions that might provide some context: (1) (2) (3) (4) Let's restrict our attention to real-values functions on an open unit interval $f:(0,1)\to\mathbb R$. There are examples $\!^{[1]}$...
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47 views

Continuous on $\{0\}$ but discontinuous at $0$

Define a function $f$ on the subset $\{0\}\cup\bigcup\left\{\left(\frac1{n+1},\frac1n\right)\middle|\ n\in\mathbb N\right\}$ of $[0,1]$ as follows: $$ f(x) = \begin{cases} 1, &\text{if $n\in\...
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Can $x^{p/q}$ be rigorously proven/disproven to be extended to a larger subset of reals that includes negative real numbers?

There have been numerous arguments that $x^{p/q}$, if $p/q$ is not an integer, should not be extended to a subset of reals that includes negative non-integers $x$-values. Many have concluded that this ...
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1answer
22 views

Difficulties on proving the continuity part of a homeomorphism

I am trying to prove that the open unitary disk $\mathbb{D}^n$ is homeomorphic to $\mathbb{R}^n$, so the way i am doing it is by showing that the function $$f(x)=\frac{1}{1-|x|}x$$ where $x$ is a ...
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Find a function $f:[0,1] \to \mathbb{R}$ that is absolutely continuous on $[\epsilon,1]$ for each $\epsilon>0$ but not on $[0,1]$.

Give an example of a function $f:[0,1] \to \mathbb{R}$ that is absolutely continuous on $[\epsilon,1]$ for each $\epsilon>0$ but not absolutely continuous on $[0,1]$. I'm thinking $f(x)= \frac{\...
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3answers
21 views

Differentiation of subtraction

I've got an exercise to do and I don't really know what to do. Exercise : We've got function $f$, where $f(a) = 0$ and $f'(a)$ exists. Also we got function $g$ which is continuous. Does exist $(f-g)'(...
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1answer
29 views

Prove that the integral of a sequence converges uniformly

I've been stuck on this problem for a bit now: Let $g$ and $f_0$ be continuous functions on $[0,1]$. Define the sequence on $[0,1]$ by $$f_n(x) = \int_0^t g(t)f_{n-1}dt.$$ I have to prove that the ...
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1answer
49 views

can we make continuous functions into smooth functions?

Is there any way, to make continuous function with some sharp edges smooth function? for example if i consider a function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x)=|x|$, this function is ...
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1answer
31 views

Continuous function on compact interval $[a,b]$ with non-negative values

let $f:[a,b]\longrightarrow[0, \infty)$ be a continuous function satisfying the following: $f(\frac{a+x}{2})+f(\frac{2b+a-x}{2})=f(x), \forall x \in [a,b]$. Then the only function that satisfies these ...
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4answers
43 views

prove that a non constant periodic, continuous function has a “smallest period”

Let $\ f:\mathbb{R}\to\mathbb{R} \ $ be a non constant, continuous and periodic function. Prove that $f$ has smallest/minimum period. The definition of period that I work with is: $p$ is a period of ...
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1answer
36 views

Show that $\Delta = \{(y,y):y\in N\}\subset N\times N$ is a closed subset of $N\times N$

I have to show that $$\Delta = \{(y,y):y\in N\}\subset N\times N$$ is a closed subset of $N\times N$ I can do this by showing that its complement is an open subset of $N\times N$, but a previous ...
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1answer
18 views

$M,N$ metric spaces, $\phi:M\to N$ a surjective open map. Show that the map $f:N\to P$ is continuous iff $f\circ \phi$ is continuous

I need to show the following: $M,N$ metric spaces, $\phi:M\to N$ a surjective open map. Show that the map $f:N\to P$ is continuous iff $f\circ \phi$ is continuous In order to show that the composite ...
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1answer
34 views

Continuity and differentiability of $f(x)=\sqrt{1-x}+\sqrt{x-1}$ [closed]

Given a function $f(x)=\sqrt{1-x}+\sqrt{x-1}$. Is $f(x)$ a continuous function. Is $f(x)$ a differentiable function.