# 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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### Continuity of $\theta(x,y) = \frac{(x,y)}{\sqrt{x^{2} + y^{2}}}$ in $\mathbb R^2 - \{ (0,0) \}$ using epsilon and delta definition

I recently tried to prove the continuity of this function $\theta(x,y) = \frac{(x,y)}{\sqrt{x^{2} + y^{2}}}$ in $\mathbb R^{2} - \{ (0,0) \}$ using the inverse image of some open in $\mathbb R^2$ but ...
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### How to prove that the convergence of $(f_n)$ is uniform on compact sets?

I'm reading Theorem 0.9 in this lecture note. Below is my attempt where I got stuck at the end. Could you elaborate on how to finish the proof? Let $C$ be an open convex subset of a Banach space $X$....
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### Showing that $f((x_1,x_2),(y_1,y_2)):=x_1-y_1$ is continuous.

Define $f:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}$ by $$f((x_1,x_2),(y_1,y_2)):=x_1-y_1$$ I want to show that $f$ is continuous. I already know that the function $g(x,y):=x-y$ is ...
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### What is range of values that the word 'nearby' supposed to represent in this informal definition of continuity.

In my book it gave two informal explanation for the concept of continuity. I had doubt in the second explanation but I cleared it by asking it here. The explanation is , Suppose a function f has the ...
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### In this informal definition of continuity (at $x=p$), what does "regardless of the manner in which $x$ approaches $p$" mean?

In my book it gave the informal definition of continuity as If we let $x$ move toward $p$, we want the corresponding function values $f(x)$ to become arbitrarily close to $f(p)$, regardless of the ...
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### Find a discontinuous linear transformation from the set of bounded real sequences to the reals

I am looking for a linear transformation T: B(N,R) -> R, with R the real numbers and B(N,R) the set of all bounded real sequences (with the sup norm), that is discontinuous. I first thought that ...
1 vote
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### What does "contained continuously" mean? As in, "metric space $X$ is contained continuously in metric space $Y$".

What we mean by the "Contained continuously" terminology?! Definition: Let $H$ be a Hilbert space and $E$ is an open bounded subset of $\mathbb[C]$. The Hilbert space of measurable ...
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### Prove the $\sin$ function is continuous at $0\in\mathbb{R}$

I need to test this and I don't understand how the first part is related to the second part. Claim Use the identity $|\sin(x)| \leq |x|$ when $0 < |x| < \pi/2$ to show that $\sin(x)$ is ...
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### Proof: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$?

Question: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$? Answer: Thank to @TonyK @Ryszard Szwarc. I think that i found an ever stronger demonstration that ...
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### Show that $M \setminus f^{-1}(c)$ is disconnected if $f:M\rightarrow \mathbb{R}$ is continuous and $c \in (\min f(M), \max f(M))$.

Let $f : M \rightarrow \mathbb{R}$ be a continuous function. Prove that if $c \in \mathbb{R}$ is strictly between minimum and maximum of $f$ in $M$ then $M \setminus f^{-1}(c)$ is disconnected. Can ...
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### Must a certain continuous map have 0 in its image, given that its restriction to the unit sphere is homotopic to the identity?

Suppose $f:\mathbb{B}^n \to \mathbb{R}^n$ is continuous (here $\mathbb{B}^n$ refers to the $n$-dimensional unit ball). Suppose also that its restriction $g := f|_{\mathbb{S}^{n-1}}$ does not have $0$ ...
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### Does finiteness of the definite integral over a function imply continuity? [closed]

For example, if \begin{align*} \int_{a}^{b} f(x)\mathrm{d}x = C \end{align*} for some finite real number $C$, is $f:[a,b]\rightarrow\mathbb{R}$ continuous on $[a,b]$ ? If not, is there any further ...
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### Ideals of $C^0([0, 1]; \Bbb R)$ and compactness [duplicate]

Let $C := C^0([0, 1]; \Bbb R)$ the ring of continuous real functions on $[0, 1]$. Let $I \subset C$ an ideal. We suppose that $I$ is not contained in any $I_x:= \{f \in C \lvert f(x) = 0\}$. Show that ...
### Existence of a function $f:\mathbb{R}^8\to \mathbb{R}$ [closed]
Write $(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)$ for points in $\mathbb{R}^8$. Determine if there exist a function $f:\mathbb{R}^8\to \mathbb{R}$ satisfying all of the following: $f$ is continuous and ...
### Does this implies $(X, \tau_X)$ discrete space?
$(X, \tau_X)$ and $(Y, \tau_Y)$ be two topological space such that $card(\tau_Y)\ge 3$ $f\in C(X, Y) \quad , \forall f \in Y^X$. Does this implies $(X, \tau_X)$ discrete space? I know if the ...