Questions tagged [continuity]

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) or (general-topology)

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. A map $f : X \to Y$ is said to be continuous at $x_0$ if for every $\varepsilon > 0$, there is $\delta > 0$ such that $d_Y(f(x_0), f(x)) < \delta$ whenever $d_X(x_0, x) < \varepsilon$. A map $f : X \to Y$ is said to be continuous if it is continuous at $x_0$ for all $x_0 \in X$.

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces. A map $f : X \to Y$ is said to be continuous if $U \in \tau_Y$ implies that $f^{-1}(U) \in \tau_X$.

In the case of metric spaces, the metric induces a topology, and the two notions of continuity coincide. Note that multiple metrics can induce the same topology, and that not all topologies are metrizable (can be generated from some metric).

Continuity is a sufficient condition for the intermediate value theore. It is necessary for the extreme value theorem, as well as differentiability.

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Set of Discontinuities of arbitrary function is F sigma set

I am trying to prove that, given an arbitrary function $f$, the set of discontinuities of $f$, denoted $D_f=[{x\in{\Bbb{R}}:\mbox{f is not continuous at $x$}}$], is a $F_\sigma$ set - that is, a countable union of closed sets. This question may have…
Daniele1234
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How do you prove that $x|x|$ is differentiable at all points?

In general how do you prove that if two functions are not differentiable at a point, then it is not necessary that their product is not differentiable at that point. (Which in $x|x|$ is $0$ )
Dimsum
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$f$ partial continuous $\Rightarrow$ $f$ continuous?

$\newcommand\R{\mathbb{R}}$ Let $f\colon \R^n \to \R^m$. $f$ is partial continuous in $x=(x_1,\ldots,x_n)\in \R^n$ iff $f_k\colon\R \to \R^m, x' \mapsto f(x_1,\ldots,x_{k-1},x',x_{k+1}\ldots,x_n)$ is continuous for all $1 \le k \le n$. My question:…
widdemanef
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How to prove this limit composition theorem?

If $$\displaystyle \lim_{x \rightarrow c}f(x)=l$$ and $$\displaystyle \lim_{x \rightarrow l}g(x)=L$$ and $f(x) \neq l$ in some punctured neighbourhood of c, then $\displaystyle \lim_{x \rightarrow c}g(f(x))=L$. Note that the first two…
ryang
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Prove or disprove: If $(x_n)$ is a Cauchy sequence in $X$ and $f$ is a continuous function on $X$, then $(f(x_n))$ is a Cauchy sequence.

Note that $X$ is a metric space and $f:(X,\rho)\rightarrow (X,\sigma)$. Disproving: Suppose $(x_n)$ is a Cauchy sequence, $f$ is continuous and $(f(x_n))$ is not a Cauchy sequence, then $\exists\epsilon'>0$ such that $\forall M\in\Bbb{N}$, there…
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If $(k_n)$ is a sequence in $K \subseteq \mathbb{R}$ such that $k_n \to k_0 \not\in K$, then there exists an unbounded continuous function on $K$

Here's the question: Let $K\subseteq \mathbb{R}$ and suppose that there exists a sequence $(k_n)$ in $K$ that converges to a number $k_0 \not\in K$. Show that there is an unbounded continuous function on $K$. Here's the function I choose:…
ashK
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I am using neighborhood balls to define continuity. Are these definitions of pointwise continuous and uniform continuous correct?

I seem to understand topology more than analysis and was wondering if these definitions of continuity, which to me have more a topological flavor, are correct. Suppose $X$ and $Y$ are metric spaces and let $f: X \to Y$. $f$ is pointwise continuous…
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Uniform Continuity and partial sums equation proof

Given $f$, a uniformly continuous function defined on the interval $[0,1]$, I need to prove that $$\lim_{n\rightarrow \infty} \frac{1}{2^n} \sum_{k=1}^n (-1)^k \binom{n}{k} f(k/n)=0.$$ I have tried tackling this exercise from a couple of angles…
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Continuous function on $[0,1]$, $f(0)=f(1)$

I came across this very interesting question, which seems to be partially answered in a couple posts around here: Let $f:[0,1]\rightarrow\mathbb{R}$ continuous such that $f(0)=f(1)$. Then for all $n>1\in \mathbb{N}$, there exists $x_n\in [0,1-1/n]$…
MathNewbie
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Is $f'(a)$ Not same as Finding Derivative of $f(x)$ and Substituting $x=a$

I got this doubt regarding the differentiability of function $f(x)$ at $x=0$ where $$f(x)= \begin{cases} x^2 \sin\left(\frac 1 x\right) & x \ne 0\\ 0 & x=0 \end{cases} $$ We know that $f(x)$ is Differentiable at…
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Why doesn't Dirichlet function have a derivative in X=0

$\newcommand{\dirichlet}{\mathop{\rm dirichlet}\nolimits}$ I'm trying to find two examples for the following criterias: A method that is continuous in exactly one point but doesn't have a derivative in that point A method that is continuous in…
vondip
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$f : \mathbb R \to [-2,2]$ be a twice differentiable with $f(0)^2+ f'(0)^2=85$

Let $f : \mathbb R \to [-2,2]$ be a twice differentiable with $f(0)^2+ f'(0)^2=85$. Is it true that $\exists a \in (-4,4)$ such that $f'(a) \ne 0$ and $f(a)+f''(a)=0$ ? I think I have to use IVT / MVT to some nicely constructed new function, but I…
user495643
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Right continuous and monotone function must exist right derivative?

Right continuous and monotone function must exist right derivative? Suppose $f:R\rightarrow R$ is a right continuous and monotone function, i.e. $f(x+)=f(x),\forall x\in R$ and $f(x)$ is monotone, say non-decreasing. Does the limit exist…
Ethanabc
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Cubic root epsilon delta proof

I'm reviewing this epsilon delta proof for the continuity of the cubic root: but I can't see why is so evident that $\sqrt[3]{x^2}+\sqrt[3]{xc}+\sqrt[3]{c^2}\ge \sqrt[3]{c^2}$. Any help please? Thanks in advance.
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