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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Intermediate-Value Theorem - Find roots of an equation

I've an homework question where i need to prove that the following equation contains at-least three roots $ {x^4 \over 10} = {x^4-100 \over x-1} $. I was able to find three roots after redefining ...
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Prove that if $f$ and $g$ are uniformly continuous on A and are both bounded on A, then $fg$ is uniformly continuous on A.

Let $f$ and $g$ be uniformly continuous on A. Then given $\epsilon >0$ there exists a $\delta_{1} > 0$ such that if $|x-y| < \delta_{1}, \forall x,y \in A$, then $|f(x)-f(y)| < ...
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The action of a topological group on the function space is continuous?

Sorry for my bad english. Let $X$ and $Y$ be two topological spaces, and $G$ a topological group, let $\theta : G \times X \to X$ be a continuous action of $G$ on $X$. We defined the action of $G$ on ...
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If $f: [a,b] \rightarrow [c,d]$ is a continuous bijection and $f(a)<f(b)$ then prove that for all $a<x<b$, then $f(a)<f(x)<f(b)$.

I am having trouble proving this problem. If $f: [a,b] \rightarrow [c,d]$ is a continuous bijection and $f(a)<f(b)$ then prove that for all $a<x<b$, then $f(a)<f(x)<f(b)$. I was told ...
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Differentiable function under specific topological constraints

Can you give an example of$\:\:\emptyset\neq D\large⊂$$\:\mathbb{R}$ and a differentiable function $f$ : $D → \mathbb{R}$ such that $\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:D ⊂$$Acc(D)$, ...
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How to prove that continuous function do not necessarily preserve cauchy sequences

I am trying to construct a proof that continuous function do not preserve Cauchy sequences Every proof I can find is disprove by counter example, which is great but these counter examples cannot be ...
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40 views

Convergence of continuous function

Let $f$ be a continuous function on [$a$,$b$] mapped onto $\mathbf{R}$ and which is differentiable at $c$ $\in$ ]a,b[. (i)Show that $\exists$ a unique function $\epsilon$:]$a$,$b$[ mapped onto ...
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43 views

Let $p_n = n$th odd prime. When is $p_n$ a continuous function of $n$?

Under what topologies is the function $p(n) = n$th odd prime continuous? If we take the Euclidean topology on $\Bbb{R}$ and induced it onto the subspace $\Bbb{N}$ and called it $\tau$. Then isn't ...
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Is $f(x,y) = \dfrac{x \sin(y^2)}{x^2+y^2}$ if $ (x,y) \neq (0,0)$ and $0$ if$ (x,y) = (0,0).$continuous?

I want too know wheter the folowing function is continuous or not, but I have no idea how to do this. $$f(x,y)=\begin{cases} \dfrac{x \sin(y^2)}{x^2+y^2}, &\text{if }(x,y)\neq (0,0)\\ 0, ...
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Is it really true that “if a function is discontinuous, automatically, it's not differentiable”? [duplicate]

I while back, my calculus teacher said something that I find very bothersome. I didn't have time to clarify, but he said: If a function is discontinuous, automatically, it's not differentiable. ...
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{$f$ > a} is always open if f is continuous on $R^n$?

Lef a function $f$ be defined and continuous on $R^n$, the range of $f$ is the extended real numbers. My book claims that the set $\{x\in R^n :f(x)>a\}$ is open for all $a \ \in R^n$. I am ...
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1answer
23 views

Multivariate intermediate value theorem

The following claim seems true for me: For any continuous function over reals $f(\vec{x})$, if $f(\vec{x})=c$ has no zero, then either $f(\vec{x})>c$ for all $\vec{x}$ or $f(\vec{x})<c$ for all ...
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Complex substitution allowed but changes result

It is well known that $$ I := \int_L \frac{1}{z} ~\text{d}z = 2 \pi i $$ where $L$ is the complex unit circle, parametrized by $\gamma(t) = e^{it}, 0 \leq t \leq 2 \pi$. However, using complex ...
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primitive function of f(x)

Let $f(x)=2x-\lfloor \sin x \rfloor$. I need to say if $f(x)$ has a primitive function in $[0, 2\pi]$ and to compute $F(x)=\int_{0}^{x}f(t)dt$ in $[0,2\pi].$ Now, I think $f(x)$ has no primitive ...
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a.e. continuous to left continuous

Let $f:\mathbb{R} \to \mathbb{C}$ be a $\lambda$-a.e. continuous function. Is the following statement true? There exists another function $g:\mathbb{R} \to \mathbb{C}$ such that $g$ is left ...
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Can we find a general $\delta$ to prove the continuity of polynomials?

Polynomials are continuous functions. In other words, for all $\epsilon > 0$ and all $a$, there is some $\delta > 0$ such that if $|x-a|<\delta$, $|P(x)-P(a)|<\epsilon$ where $P(x)$ is a ...
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1answer
26 views

Verifying if a function is a.e. equal to a continuous function then it is continuous a.e.

Is it true that if a function is almost everywhere equal to a continuous function then it is continuous almost everywhere? I know that the converse is false. i.e. if f is continuous a.e. , there ...
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Why can we prove that this piecewise function is continuous at $x = 0$?

I have this piecewise function $f(x) = \begin{cases} \frac{\sin(-8x)}{8x} & x < 0 \\ (2x + 9k - 7) & x \ge 0 \end{cases}$ Now the formula for calculating the limit of ...
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Functional Limits and Continuity

so here is a problem I have been working on: Assume g is defined and continuous on all of $\mathbb{R}$. If $g(x_0) > 0$ for a single point $x_0 \in \mathbb{R}$, then $g(x)$ is in fact strictly ...
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1answer
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Functions continuous at only one point - more exotic examples?

The canonical example of a function continuous at only one point is $$f(x) = \chi_{\mathbb{Q}}(x) \cdot x$$ which is continuous only at $0$. A user on another question pertaining to this issue has ...
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Separating the supports of disjoint continuous functions

Can the supports of disjoint continuous functions on a compact Haussdorf space always be separated by open sets? I.e.: given a compact Haussdorf space $X$ and a sequence of continuous functions ...
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The integral of a discontinunous function is not absolutely continuous function

We know that even if a function $f(x)$ is not a continuous function of $x$, it's integral $\int_{-\infty}^x f(s)ds$ can still be a continuous function of $x$. I have a question: how horrible (in terms ...
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If $f>0$ is continuous on a bounded closed interval, then $1/f$ is bounded there

Here's a problem I've been working on: If $f$ is continuous on $[a, b]$ with $f(x)>0$ for all $a \le x\le b$, then $\frac 1f$ is bounded on $[a, b]$. My thoughts so far: $1/f$ is bounded so long ...
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1answer
62 views

Proof of Urysohn's lemma (or what my teacher called with that name)

In class, I have been given the following statement. Urysohn's lemma Suppose $X$ is a locally compact Hasudorff topological space and $K\subseteq V\subseteq X$ are respectively a compact and an open ...
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2answers
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continuous and uniformly continuous

I remember proving $x^3$ is not uniformly continuous on $\mathbb{R}$. Then I read the proof of a theorem: Suppose $D$ is compact. Function $f: D \rightarrow \mathbb{R}$ is continuous on $D$ if and ...
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How to show that $(x,y) \to x$ is continuous?

Let $p: (x,y) \subset \mathbb{R}^2 \to x \subset \mathbb{R}$ be the projection function How do I show that $p$ is continuous? I considered using the topological definition. Let $A \subset ...
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How can I determine if this function is continuous at x=1?

Is 2/0 a discontinuity or infinity for a function? For the question: Given the function $ (x^2+1)/(x-1) $, is the function continuous at x=1? When I took the right hand and left limits, I got ...
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65 views

Joint PDF in a circle area.

I don't understand how can I solve this. My only guess it's that it's related with the probability of the circle area of c. The coordinates X and Y of a point are independent zero mean normal random ...
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general definition of concavity, mean-preserving spread and concavity

The usual definition of concavity is: for any $x_1$ and $x_2$ and any $t\in[0,1]$, $$f(tx_1+(1-t)x_2)\geqslant tf(x_1)+(1-t)f(x_2).$$ I am wondering how to generalize this definition to more than 2 ...
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Riesz transform does not preserve continuity

I've read somewhere that the Riesz operator $R_j$ defined by $$R_j f(t) := c(n) \, \text{pv} \int_{\mathbb{R}^n} \frac{x_j}{|x|^{n+1}} f(t-x) \, dx$$doesn't preserve the continuity, but I can't ...
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Looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism

I am looking for an example of an infinite metric space $X$ such that there exist a continuous bijection $f: X \to X$ which is not a homeomorphism . Please help . Thanks in advance .
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To check the continuity of $f$

Let $f:(C^1[a,b],\|.\|_{\infty})\to \mathbb K$ and $c\in (a,b)$ be such that $f(x)=x^{'}(c)$ for all $x\in C^1[a,b]$. Then is $f$ a continuous linear functional? I chose $a=0,b=1$ and $c=1$, then ...
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Differentiability class of Matern function (based on Modified Bessel Function of second kind)

I am working on some techniques using the Matérn covariance function: $h(r) = \frac{2^{1-\nu}}{\Gamma(\nu)}\Bigg(\sqrt{2\nu}\frac{r}{\rho}\Bigg)^\nu K_\nu\Bigg(\sqrt{2\nu}\frac{r}{\rho}\Bigg)$ with ...
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Let $A\subset \Bbb R^2 $ with the property that every continuous function on $A$ has a maximum in $A$ .Prove that $A$ is compact.

Let $A\subset \Bbb R^2 $ with the property that every continuous function on $A$ has a maximum in $A$ .Prove that $A$ is compact. My try: We have to show that $A$ is closed and bounded. In order to ...
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Continuity of the directional derivatives implies continuity at the point ?

This might be a trivial question. Consider a function $f:\mathbb{R^2}\rightarrow \mathbb{R}$ and consider some point $(a,b)\in \mathbb{R^2}$. Suppose we know that all the directional derivatives ...
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1answer
30 views

Show that $f$ is differentiable at $x=1$.

Let $f$ be a real valued continuous function defined on $[0,2]$ such that $f$ is differentiable at all point except possibly at $1$. Suppose that $\lim_{x\to 1}f^{'}(x)=5.$ Show that $f$ is ...
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Does differentiability on a set imply continuous differentiability on the set? Counterexample?

Of course, differentiability implies continuity, but for a function to be differentiable on a set, say $[a,b]$, then, for the limit to exist, would we not need it to be defined on the set? I hear ...
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An everywhere discontinuous function

As usual, $\mathbb R[x]$ denotes the vector space of polynomials in one variable with real coefficients. It is easy enough (and a good exercise for beginners) to prove that the function ...
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Prove the following functions are not continuous at c?

for $c = 1$ $$f(x) = \begin{cases} x^2-1, & \text{0 < x ≤ 1} \\ x+3, & \text{x > 1} \end{cases}$$ My solution: Function $f$ is defined at $x=1$ since $f(1)= 1^2-1 = 0$ the ...
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Riemann type function continuity

Define real-valued function $f$ on $\mathbb{R}\cap[0,1]$ by setting $$f(x)= \begin{cases} x,\,\,\text{if $x$ irrational}\\ p\sin(\frac{1}{q}),\,\, \text{if $x=\frac{p}{q},\gcd(p,q)=1$}\\ ...
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Suppose that a density function is given by the formula $f(x)$. Find the probability that $x$ is between $8$ and $14$.

I am trying to figure out the following problem presented in my probability homework. Suppose that a density function is given by the formula $$f(x) = \begin{cases} \frac{2}{108}x & \text{if} ...
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1answer
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Is there a name for this property of maps between topological spaces?

Let $X$, $Y$ be Hausdorff locally compact spaces and let $f \colon X \to Y$ be a proper continuous map. Consider the following property (P): $$ \text{for any compact set $K$ in $Y$ and for any open ...
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Approximation of continuous functions

Let $f$ $\in$ C([0,1]), $f(0)=0$ and $\epsilon > 0$. Prove there exists a polynomial $p$ such that $p(0)=f(0)=0$, $p´(0)=0$ and $||p-f|| < \epsilon$ . The norm is sup-norm I Know that by ...
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In general does lim |f(x,y)| = |lim f(x,y)|?

I understand that for any continuous functions f and g, the following holds: $$\lim_{x \rightarrow a} f(g(x)) = f(\lim_{x \rightarrow a} g(x))$$ For this purpose of evaluating limits, is the ...
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Problem concerning pointwise and uniform continuity

I have this problem I'm working on: Suppose $(f_n)_n$ is a sequence of functions $f_n: \mathbb{R} \rightarrow \mathbb{R}$ that converges pointwise to a function $f: \mathbb{R} \rightarrow ...
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25 views

$X$ is locally compact then $Y$ is locally compact

Let $p:X\rightarrow Y$ be a closed continuous surjective map such that $p^{-1}(y)$ is compact for each $y\in Y$. Show that if $X$ is locally compact then $Y$ is locally compact. Let $y\in Y$ and ...
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What is the difference between $\implies$ and $\land$ in definition of continuity

Can someone please explain to me what would happen if instead: Given $D \subset \mathbb{R}$ $\forall \epsilon > 0, \exists \delta$ such that $\forall x, x_o \in D, |x-x_o| < \delta \implies ...
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1answer
32 views

Prove $f$ is continuous with respect to the induced metric

$X$ is a normed vector space with norm $f: X \rightarrow \Bbb R$. Prove that $f$ is continuous w.r.t the induced metric, namely $d(x,y) = f(x-y)$. I already proved by triangular inequality that ...
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If $(M_t)_{t \in [a,b]}$ is a martingale, then $t \mapsto E [ M_t ]$ is continuous.

Suppose $(M_t)_{t \in [a,b]}$ is a stochastic process. Denote $(\mathcal{F}_t)_{t \in[a,b]}$ to be the natural filtration generated by the process $(M_t)_{t \in [a,b]}$. Moreover, suppose $(M_t)_{t ...
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how do i prove that these sets are the same? [duplicate]

Let (M,d) be a metric space, with $E\subset M$: And for this one; should i take distance from an element to itself or something? prove that: b)$\{ x: d(x,E)=0\} = \overline E$