# 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)

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### Defining matrix function through its diagonalization

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a smooth function. Let $Dg(n)$ be the set of diagonal $n \times n$ matrices, and define $\overline{f}:Dg(n) \to Dg(n)$ in the following way: first, for a ...
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### Possible values of the integral of the function with the information of derivative

Let $f: [0,1] \to \mathbb{R}$ denote some continuously differentiable function such that $f '(x)$ is continuous and $\int_0^1 f(x)dx=0$. If $\max_{x\in[0,1]}|f'(x)|=24$, then what are some/all of the ...
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### prove that the function $f(x) = x + \frac{1}{{x^2}}$ is continuous on the interval $(2, \infty)$ using epsilon delta definitions

Let $f(x) = x + \frac{1}{{x^2}}$ and I want to prove that $f$ is continuous at a point $c$ in the interval $(2, \infty).$ I started by defining $\delta$ as $\min(1, \text{ })$ and then I tried to ...
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### Reconciling Continuity of Binary Relations with Continuity of Functions/Correspondences

I asked this question in the Economics StackExchange as well, but figured it may be better-suited here. There are various ways to express the concept of continuity of a binary relation, but one I've ...
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### $f(x) \ge 0$ for all $x \in [a, b]$ and $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$.

Suppose that $f$ is continuous on $[a, b]$, that $f(x) \ge 0$ for all $x \in [a, b]$ and that $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$. My attempt: Let $\dot{\Pi}$ be a tagged ...
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### Showing a certain type of function on $\mathbb R ^d$ is Lipschitz?

I want to prove the following: Assume that $f:\mathbb R ^d \to \mathbb R$ is continuous, convex and $|f(x)|\leq a+b|x|$. Then $f$ is Lipschitz. I thought it would follow immediately from the ...
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### Show that the expression defines a norm

Let us consider the real vector space $V := C([-1, 1])$. Show that the expression $|| f || := max_{x ∈ [-1,1]}abs(\frac{1}{2}f(x))$ (for any $f ∈ V$) defines a norm on $V$. To show that the expression ...
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### How can I prove that if $f:[0,+\infty)\to[0,1]$ is continuous and increasing, then it is uniformly continuous? [closed]

The problem is the following: Let $f:[0,+\infty)\rightarrow\mathbb{R}$ such that it is continuous, strictly growing and its image is contained in $[0,1]$, is uniformly continuous. I've tried in ...
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### Existence of all partial derivatives implies anything about total derivative’s existence?

Say a function $f: \mathbb{R}^n \to \mathbb{R}^m$ is $C^{\infty}$ if its partial derivatives of all orders exist. Does this imply anything about the existence of its total derivatives, or if not then ...
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### Semidifferentiability at the extremum of an interval and continuous extension of derivative

Let $f:[a,b] \to \mathbb{R}$ be differentiable on $(a,b)$ with continuous derivative $f'$. (i) Assuming that $f'$ can be continuously extended at $a$, is it true that $f$ is semidifferentiable at $a$ ...
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### Show that there does not exist any holomorphic function on the open unit disk and continuous on the closed unit disk with the given property.

Let $\mathbb D : = \left \{z \in \mathbb C\ :\ \left \lvert z \right \rvert < 1 \right \}.$ Prove that there is no continuous function $f : \overline {\mathbb D} \longrightarrow \mathbb C$ such ...
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### Mistake in solution to show that $\frac{x^ay}{x^2+y^2}$ is continuous at $(0,0)$ if $a>1$.

I was working through some practice problems, and I'm not sure where I'm wrong in the following proof. The function $g:\mathbb{R}^2 \rightarrow \mathbb{R}$ is defined by \begin{cases} \frac{x^ay}...
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### A question regarding the deformation retraction of a ball without origin to a sphere.

Let $D=\{||x||\leq 1\}\subset \mathbb R^d$ be a ball or radius $1$ with center at origin $O$ and let $H:\overline (D\setminus\{O\})\times[0,1]\to D\setminus\{O\}$ be a strong deformation retraction to ...
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### Dominated Convergence Theorem and Norm convergence

Here is the theory I have in mind: Dominated Convergence Theorem: $(X,A,\mu)$ measure space, $g$ a $[0,\infty]$-valued integrable function on $X$, $f,f_1,f_2,\dots$ $[-\infty,\infty]$-valued $A$-...
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### Translation in $L^{\infty}$

Consider the translation operator $\tau_h$ defined on $L^\infty(\mathbb{R}^n)$ s.t. $\tau_hu(x)=u(x-h)$. I know that $\tau_h$ is not continuous with respect to $h$, I mean it’s not true that $h\to 0$ ...
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### Suppose that exists a $M>0$ s.t. $|f(x)-f(p)| \leq M|g(x)-g(p)|$ for all $x$. Prove that if $g$ is continuous at $p$, then $f$ is as well.
"Let $f$ and $g$ be defined in $\mathbb{R}$ and suppose that exists a $M>0 \hspace{0.2cm}(\exists)$ st. $|f(x)-f(p)| \leq M|g(x)-g(p)|$ for all $x$. Prove that if $g$ is continuous at $p$, ...