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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Continuity at $+\infty$ for the function defined by $f(0)=\infty$ and $f(x)=1/x$ for $x \in (0,10]$.

Let the domain of the function $f(x)$ be $[0, 10]$, and its range be the extended real numbers (including +$\infty$ and -$\infty$). Define: $ f(x) = \left\{ \begin{array}{lr} 1/x & ...
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36 views

Inner Product on Linear Space of Continuous Functions

Consider the linear space of continuous functions $C[-1,+1]$ defined over the interval $[-1,+1]$. We define an inner product $\langle\cdot , \cdot\rangle$ on $C[-1,+1]$ by $$\langle ...
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1answer
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How to prove continuity in Baire space?

Let $X=(\omega^\omega,d)$ be Baire space with the metric $d$ defined in assignment $1$. Define a function $G:X\to X$ by letting, for $f\in X$, the function $G(f)$ be defined by: ...
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1answer
56 views

What's equicontinuous? What's uniform equicontinuous? What's pointwise equicontinuous?

I have some problem about my homework. Is the sequence of function $f_n:\mathbb R \to \Bbb R$ defined by $$f_n(x) = cos(n+x) + log(1+\frac{1}{\sqrt{n+2}}sin^2(n^nx))$$equicountinuous? ...
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1answer
38 views

If $f(x)$ is differentiable on $(a,b)$, is $f(x)$ continuous on $(a,b)$? [closed]

Let $f(x)$ is a differentiable function on $(a,b)$. $f$ is continuous on $[a,b]$ - False - Counterexample $1/x$ Now I should prove that $f$ is continuous on $(a,b)$ I know that $f'$ exists by the ...
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41 views

Is the function $f:(a,b) \to \mathbb R$ defined by $f(x):=\dfrac {x-(a+b)/2}{(x-a)(b-x)} , \forall x \in (a,b)$ a homeomorphism?

Is the function $f:(a,b) \to \mathbb R$ defined by $$f(x):=\dfrac {x-\dfrac{a+b}{2}}{(x-a)(b-x)} , \forall x \in (a,b)$$ a homeomorphism ? I have noticed that it is continuous and also noticed that ...
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1answer
27 views

Given a limit of $f$, find $f(1)$ and $f'(1)$.

Suppose $f$ is a continuous function, and $$ \lim_{h \to 0} \frac{f(1+h)}{h} = 5 $$ Find $f(1)$. Find $f'(1)$.
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4answers
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Remark 4.31 in Baby Rudin: How to verify these points?

Let $a$ and $b$ be two real numbers such that $a < b$, let $E$ be any countable subset of the open interval $(a,b)$, and let the elements of $E$ be arranged in a sequence $$x_1, x_2, x_3, ...
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1answer
42 views

If $f$ is absolutely continuous on $[0,1]$, show $f(A) = \{f(x) : x\in A\}$ has Lebesgue measure 0

Suppose $f$ is absolutely continuous on $[0,1]$, and for $A \subset [0,1]$, we let $f(A) = \{f(x) : x \in A\}$. Prove that if $A$ has Lebesgue measure $0$, then $f(A)$ has Lebesgue measure $0$. My ...
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Need help proving $h$ is continuous at $0$

Here's the problem: Define $h: \mathbb{R} \to \mathbb{R}$ by Prove $h$ is continuous at $0$. Usually, I have a good idea how to prove this when the cases are $x=0$ and $x\neq0$, but I'm a bit ...
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1answer
41 views

Is $(\mathbb S^3 \setminus \{0,0,0,1\}) \cap \mathbb R^3 $ homeomorphic with $\mathbb S^2 \times \mathbb R $ ?

Is $(\mathbb S^3 \setminus \{0,0,0,1\}) \cap (\mathbb R^3 \times \{0\})$ homeomorphic with $\mathbb S^2 \times (\mathbb R \times \{0\}\times\{0\})$ ?; here by $\mathbb R^3 \times \{0\}$ I mean ...
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1answer
31 views

Can we prove, without invoking invariance of domain, that $\mathbb R$ and $\mathbb R^2$ are not homeomorphic?

Can we prove, without invoking invariance of domain, that $\mathbb R$ and $\mathbb R^2$ are not homeomorphic, or equivalently, that no open set of $\mathbb R$ is homeomorphic to an open set of ...
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1answer
36 views

Existence of continuous bijective function $f:[0,1] \times [0,1] \to [0,1] $ ? Continuous and only injective and continuous and olny surjective?

Does there exist any continuous bijective function $f:[0,1] \times [0,1] \to [0,1] $ , where $[0,1]$ is equipped with usual Euclidean metric of $\mathbb R$ and $[0,1] \times [0,1]$ is equipped with ...
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Hölder continuity of $\frac1x$

I have a question. Is the function $f(x)=1/x$ Hölder continuous if $x\in (\varepsilon,+\infty),\ \varepsilon>0$?
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2answers
32 views

Proof: f total differentiable then f continuous

I'd like to show that if $f: O \subseteq \mathbb{R} n \to \mathbb{R}m$ is differentiable in $x_o \in O$, then $f$ is continuous in $x_o$. My idea: If $f$ is (total) differentiable in $x_o$ then ...
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2answers
46 views

how to understand the definition of continuity in analysis?

Please have a look at the picture above. This is about the continuity in analysis. I don't really understand how to utilize this definition? It says that is statement is equivalent to f is ...
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0answers
26 views

Approximation of continuous functions by polynomials

Let $D$ be a compact set in $R^d$. Assume $f \in C(D)$ is such that $\int_D f(x) x^\alpha dx = 0$ for any multi-index $\alpha = (\alpha_1, ... \alpha_d)$. Show that $f(x) = 0$ for all $x ∈ D$. ...
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1answer
21 views

Continuity of a rational function

I have to evaluate the continuity of a two functions at a couple of different points and I am a bit stuck. Here are the two functions: $f(x,y) = 0$ if $(x,y)=(0,0)$, $f(x,y) = ...
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1answer
19 views

Limits in a closed interval

I have the following result: $$\forall c\in S \,\,\, \lim_{x \rightarrow 0} F(c+x) = F(c) $$ Using a theorem prover, I can prove the result if I assume: $F$ is continuous on $S$ The interval $S$ ...
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1answer
39 views

Continuous bijection of two balls

Let $X$ be the union of the open unit ball and its bourdary on the first quadrant. Let $Y$ be the union of the open unit ball and its bourdary on the first and second quadrant. Does there exists a ...
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Proof strategy: continuity of an integral

Consider $g: I \rightarrow \mathbb{R}$ given by $$g(x) = \int_{x_0}^x f(y)dy$$ If $f : I \rightarrow \mathbb{R}$ is Riemann-integrable, then $g$ is continuous on $I$. Proof: For any $x_1 \in I$, ...
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1answer
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When does continuous (or uniformly continuous ) function between normed linear spaces carries bounded sets to bounded sets ?

I know that if $f:\mathbb R^m \to \mathbb R^n$ is continuous then $f$ carries bounded sets to bounded sets . What if we say $X,Y$ are normed linear spaces and $f:X \to Y$ where $f$ is continuous ? ...
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136 views

If $f:X \to X$ is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous?

If $f:X \to X$ (codomain and domain have the same topology) is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous? Note that the orbit being finite and $f$ being a ...
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1answer
38 views

Proof of continuity of all functions on N

My task is to basically proof that any function defined on $\mathbb N$ is a continuous function. I wanted to use the definition that states that if $f$ is continuos at every point a in the domain ...
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2answers
22 views

Example is required

I am trying to find a seuqence of a continuous functions $\{f_n\}$ defined on $[0,1]$ bounded by some small number, say $\varepsilon$ with the additional requirement of $f_n^\prime(t_0)=1$ at a ...
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The definition of $C(\bar U)$.

In 'Partial differential equations, Evans', $C(\bar U)$ is defined by the space of continuous functions $u\in C(U)$ such that $U$ is uniformly continuous on bounded subsets of $U$. But I have known ...
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Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] ...
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How do I prove or disprove $h$ is continuous at $0$? [duplicate]

I did a problem similar to this where we use a sequence ($x_n$) = $\frac{1}{\sqrt{2n\pi + \frac{\pi}{2}}}$. And then we take the limit? Can someone help me out here?
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1answer
23 views

What is K-Lipschitz and how do I use it to prove this problem?

So, I'm completely lost here. Can someone explain to me what K-Lipshitz is? And how I'm supposed to prove this problem?
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Connection between Darboux property and semicontinuity

Is there a connection between the Darboux property (that is, the mean value propriety) and semicontinuity? That is, is there a characterization of semicontinuity that uses Darboux property (or are ...
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1answer
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Let $f:\mathbb{R}→\mathbb{R}$ be defined by $f(x)=\sqrt[3]{x}$. Use the definition of continuity to prove that $f$ is continuous at 0?

Let $f:\mathbb{R}→\mathbb{R}$ be defined by $f(x)=\sqrt[3]{x}$. Use the definition of continuity to prove that $f$ is continuous at $0$? How am I supposed to do this? Can someone please help ...
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31 views

If a function is continuous at $x_0$, then it must be defined on a neighborhood of $x_0$?

The title is all my question. Let me state it again: If a function is continuous and defined at $x_0$, then it must be defined on a neighborhood of $x_0$? It seems trivial, but I cannot prove ...
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1answer
40 views

If a continuous function is never $0$, it must either be always positive or always negative

Let $I$ be an interval, and let $f : I \to \mathbb{R}$ be a continuous function on $I$. Suppose $f$ is never $0$ on $I$. Prove that $f$ must either be always positive or always negative. I was ...
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5answers
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Is continuity the same as domain being all real numbers?

Basically, my question is— Is this statement: f(x) is continuous for all x. the same as? The domain of f(x) is all real numbers.
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1answer
22 views

Continuity and maxima and minima [closed]

Is $\sin (t)/t$ continuous at $t=0$? And also, if a function $f(x)$ is of indeterminate form at $x=a$, can it be continuous if $f(a)$ does not exist? Can a discontinuous function have a local maximum ...
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1answer
42 views

Continuity of a function on the complement of a set of Jordan measure zero

Let $f:D\subset \mathbb R^n \to \mathbb R$ $f= \begin{cases} c, & \text{if $\vec x \in \Omega$} \\[2ex] 0, & \text{if $\vec x \notin \Omega$} \end{cases}$ where D is a closed rectangle and ...
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2answers
28 views

Computing the limit of an integral (Derivatives of Integrals)

Assuming that $f(x)$ is continuous in the neighborhood of $a$, compute $$ \lim_{x \to a} \frac{x}{x-a} \int_a^x f(t)dt $$
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1answer
25 views

Difference between definition of a limit if a function and definition of continuity.

I am having trouble understanding why this particular difference exists between the definition of a limit of a function and definition of continuity. Heres the definition of a limit of a sequence. ...
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Multivariable: Continuity of Piecewise function

I have this Multivariable problem.... Where I have to find out if a function is continuous or not. Here is the problem: $f(x, y)=\left\{\begin{matrix} \frac{x^4+3y^4}{x^2+y^2} & (x,y)\neq (0, ...
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1answer
29 views

Use the definition of differentiation on a piecewise function.

I need to find the derivative at $x=0$. $$ f(x)= \begin{cases} x^2\sin(1/x) & \text{if } x\neq 0 \\ 0 & \text{if } x \leqslant 0 \end{cases} $$ Using the definition, I know that it's equal ...
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1answer
39 views

If $f:\mathbb C \to \mathbb C$ is continuous at a point $z_0$, then show that $\overline {f(\bar z)}$ is also continuous at $z_0$.

If $f:\mathbb C \to \mathbb C$ is continuous at a point $z_0$, then show that $\overline {f(\bar z)}$ is also continuous at $z_0$. Is the same true for the differentiability at $z_0$? I'm trying to ...
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2answers
26 views

Continuity and integrability;is it true?

If we have a discontinuous real function of all nonnegative terms and $\int^b_a fdx=0$, then does that necessarily imply $f(x)=0$? I can't come up with an example to help me understand.
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1answer
36 views

Discontinuity and differentiation;is this possible?

If $ f $ is a continuous function defined on a real interval that has a discontinuity at a point (but is continuous otherwise), then is it possible to be differentiable at that point?
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21 views

Rotations of the Plane

When is a rotation of a plane not continuous? I know that if I take a point $(x,y)$ then the rotation is $(x\cos(\theta)-y\sin(\theta),x\cos(\theta)+y\sin(\theta))$, but I keep getting that this ...
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31 views

Can we find such a monotone function? [duplicate]

Can we find a monotone function $f:[0,1]\rightarrow\mathbb R$ whose discontinuity set is exactly the set $\mathbb Q\cap [0,1]$? Or can we prove that such a function does not exist?
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Does having infinite limits at a point of discontinuity imply having a vertical asymptote?

Considering that discontinuities occur at holes, jumps, and vertical asymptotes. Is it possible for a function to have a limit from the left of infinite and the limit from the right - infinite if the ...
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1answer
39 views

Compute the righthand limit; calculus

Let $ f $ be a function defined on a real interval from $0 $ to $1$ and have a discontinuity at $1/2$ (however the righthand and lefthand limits still exist). Let $ F $ be a function defined by $ ...
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0answers
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Continuity of Product Topology [duplicate]

Let $X_1, X_2, Y$ be topological spaces and let $X_1 \times X_2$ be the topological space obtained by furnishing the Cartesian product set with the product topology. Let $f: X_1 \times X_2 \to Y$ be a ...
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3answers
44 views

Prove that a function that maps a discrete metric space to any metric space is continuous [closed]

Let $f:D→M$ where $M$ can be any metric space and $D$ is any set with the discrete metric. Prove that $f$ is continuous. I'm not sure where to begin with this.
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1answer
405 views

Given two potatoes, prove that there is a loop of wire which fits around both

This is a classic problem in geometric continuity and I want to see if there are some solutions other than the one I'm thinking of: Two potatoes are given. Prove that there exists a closed loop of ...