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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$F=\{x\in M: f_\lambda(x)\ge 0, \forall \lambda\in L\}$ is a closed subset of $M$ [closed]

I want to discuss this proof that: Let $f_\lambda$ be a family of continuous functions, then: $F=\{x\in M: f_\lambda(x)\ge 0, \forall \lambda\in L\}$ is a closed subset of $M$: Since every ...
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23 views

$M=A\cup B$, $f|_A$ and $f|_B$ are continuous, then $f$ is continuous in $A\cap B$

In order to prove: $M=A\cup B$, $f|_A$ and $f|_B$ are continuous, then $f$ is continuous in $A\cap B$ does it suffice to prove: for $a\in A\cap B$: since $f|_A$ is continuous, then $\forall ...
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Prove $g(y) = \int_{\mathbb{R}} \sin(y^2x)f(x) dx$ is bounded and continuous on $\mathbb{R}$ for $f \in L^1(\mathbb{R})$

This question is from a practice qualifying exam. Here's my attempt (I'm a bit stuck on the continuity part): Since $f \in L^1(\mathbb{R})$, $f$ is bounded. Then: $$|g(y)| = |\int_{\mathbb{R}} ...
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Continuity in terms of interior of preimage and preimage of interior

Let $f$ be a map between metrix spaces $X,Y$. In order to prove: $f$ is continuous $\iff$ $f^{-1}(\operatorname{Int} Y)\subset \operatorname{Int}(f^{-1}(Y))$ I did: $\rightarrow$ Suppose $x\in ...
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49 views

How to find the set of values $S$ where $f$ is not differentiable?

Let's assume we are given an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and for the purposes of this question, let's assume we know nothing about the differentiability of $f$, i.e. we have no ...
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29 views

Proving equivalence of statements on continuity between metric spaces

On page 228 of Mícheál Ó Searcóid's Metric Spaces (2007), he writes Criteria for Comparability of Metrics Suppose $X$ is a set and $d$ and $e$ are metrics on $X$. Then the following ...
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Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a ...
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1answer
39 views

$t\mapsto\sin(tA)$ is continuous

How to show that $t\mapsto\sin(tA)$ is continuous for a real matrix $A\in Mat(n,n)$ Can I use trigonometric identity, $\sin y-\sin x=2\cos\left(\frac{x+y}{2}\right)\sin(y-x)$ but this holds ...
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122 views

Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the ...
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21 views

Can you use level sets to suggest discontinuity?

Consider the function $$f(x, y) = \frac{x^2 + y^2}{y}$$ for which I already showed that the level set of height $c$ is given by a circle of the form $$C_c: (x - 0)^2 + (y - \frac{c}{2})^2 = ...
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Removing Discontinuity in 3-space without changing the partial derivative

Is it possible to find a version of the function $$f(x,y) = x\cdot \lfloor y \rfloor + \lfloor x\rfloor^2$$ That is continuous. ANY operation is allowed in changing the function as long as the ...
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39 views

Simple example of a mapping between topological spaces

I read the definition of a continuous function between topological spaces a lot of times, but I'm having difficulties to apply it to a simple example. Given two topological spaces $(X,\tau_1)$ and ...
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+50

On an injective ring homomorphism from the ring of continuous functions to the ring of differentiable functions

Let $\phi : C \to D$ be an injective ring homomorphism such that $\phi(1)=1$, where $1$ denotes the constant function $1$ and $C,D$ are the rings of continuous and differentiable functions on ...
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27 views

Obtaining a bound on the Bernstein approximation of Lipschitz functions

I encountered the following excercise in a book: Exercise: Given a Lipschitz continuous function $f$ on $[0,1]$, with Lipschitz constant $c$. Show that $|B_{n,f}(p) - f(p)|\le ...
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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
38 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 ...
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49 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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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 ...
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45 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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$f( \phi^{-1}(x_0 +h)) = f(\phi^{-1}(x_0+h))+h \alpha+ O(h^2)$ - 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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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
31 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 ...
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26 views

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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42 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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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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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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60 views

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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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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47 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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37 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
22 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 $f(x,.)\colon\mathbb{R}\to \mathbb{R},$ ...
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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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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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70 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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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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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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59 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 ...
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37 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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0answers
20 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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33 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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42 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 ...
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1answer
50 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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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 ...