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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Proving the continuity of functions from metrics [on hold]

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $(X,\rho)$ and $(X,\bar\rho)$ be metric spaces, ...
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Finding the limit of complex function

I am trying to check the continuity of this complex function at the origin. $f(z)=\begin{cases} \operatorname{Im}( \frac{z}{1+|z|} ) \qquad &\mbox{when } z\neq0,\\ 0 \qquad ...
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Real Analysis: Continuity

$f(x)=\left\{ x^2+x, x \in \Bbb Q\right\}, f(x)=\left\{ x^3 + 1, x \notin \Bbb Q \right\}$ I want to prove that $f$ is discontinuous at $x \ne 1$. What I have so far is: Fix $\delta > 0$. We ...
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finding the continuity of a function

I need to find the value of $a$ for which the function $f(x,y)= \frac{x^2-y^2}{x^2+y^2}$ if $(x,y) \neq (0,0) $ and $f(x,y)=a$ when $(x,y)=(0,0)$ when continuous along the path $y=b\sqrt{x}$ where ...
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Need help in understanding proof of continuity of monotone function

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.) Proposition: Let $A$ be ...
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a continuous function

Let $C([a,b])$ be the collection of all functions $f:\mathbb{R} \to \mathbb{R}$ such that continuous on $[a,b]$. It is known that if $f\in C([a,b])$ then $f$ is continuous on every sub-interval of ...
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If a continuous function is strictly decreasing before a point and strictly increasing afterwards, is the point a global minimum?

I'm in the middle of a proof that a point on a function is a global minimum. Usually I'd just solve an inequality to prove by contradiction that there are no points less than the minimum. But I can't ...
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Continuity of the joint distribution function given continuity of marginals

Suppose $X$ and $Y$ are continuous random variables such that $F_X$ and $F_Y$ are the respective distribution functions. Suppose $F_X$ is continuous at $x_0$ and $F_Y$ is continuous at $y_0$. Then ...
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25 views

Smooth function conditions

A curve defined by $x=f(t)$, $y=g(t)$ is smooth if $f′(x)$ and $g′(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
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Is $f\colon Y'\to Y$ continuous?

Consider $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and $T\colon X\to X$ continuous, describing the following dynamics: For $\eta\in X$ let $\eta(y)$ describe the y-th position in the bi-infite sequence ...
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Help Understanding Gradients

I understand that gradients are vectors with partial derivatives as components when working in 3D space, but does the the existence of a gradient at a point imply continuity at that point?
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Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
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Identifying a subclass of the class of monotonic transformations

Let $u$ be a continuous function from $R$ to $R$. Then $v$ is called a positive monotonic transformation of $u$ if $u(x) < u(y)$ if and only if $v(x)<v(y)$ and similarly for greater than and ...
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Why does continuity let you interchange operators?

This is one of those "dumb" questions. When solving a problem recently I found that the key to solving it was to interchange the limit operator and the exponential operator. Because the function ...
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Use the Intermediate Value Theorem to show the equation

Use the intermediate value theorem to show that the equation, $ tan(x) = 2x $ has an infinite amount of real solutions. So far I have used the IVT to show that for $ f(x) = tan(x) $ in the interval ...
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1answer
34 views

Is $H(\theta) = \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos (2\pi n_k \theta)$ for a given sequence $n_k$ equal a.e. to a continuous function?

I am studying Furstenberg's article Strict ergodicty and transformation of the torus and I'm stuck with the following construction. Define sequence $(v_k)_{k \in \mathbb{N}}$ as $v_1 =1, ...
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Definition of equi-absolute continuity

Could someone provide (or point me to) a definition of equi-absolute continuity for functions defined on an open bounded subset $\Omega \subseteq\mathbb{R}^n$? I only managed to find a definition for ...
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81 views

Proving there is no continuous function $f: [0, 1] \rightarrow \mathbb R$ that is onto

I know that onto means for every $y \in Y$ there is an $x \in X$ s.t. $f(x) = Y.$ In this case we are saying there is no continuous function that exist that where for every $y \in\mathbb R$ there is ...
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How much does convolution with a compact C^m kernel increase the order of continuity.

Let $f \in C^n$ and $g \in C^m$, with $g$ compactly supported and integrable. How much does the convolution $f\star g$ of $f$ with $g$ increase the order of continuity? Statement: I think that, under ...
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Uniqueness of a solution to an IVP over a large domain

I get that both $x_1(t)$ and $x_2(t)$ are continuous on $I$ as they are differentiable on $I$ as they are solutions to the differential equation. However I do not understand why their continuity ...
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Did I just prove $h^{-1}(W)$ is open in $X$?

Let $h:X \rightarrow Y$ be a function between topological spaces. Let $U$ be a closed subset of $X$ and $g=h|_{U}$ be the restriction. Suppose further that $g$ is continuous. Let $W$ be open in $Y$. ...
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Spivak's Calculus, Chapter 6 problem 16 d)

I don't understand how to solve this problem and the official solution does not make much sense to me either. The problem is: (d) Let $f$ be a function with the property that every point of ...
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Definition of continuity implies a discontinuous function is continuous?

So I have a text that defines a function $f$ to be continuous if $f^{-1}(A)$ is open whenever $A$ is open. However, that seems like a confusing definition since it doesn't specify if the open sets ...
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$f$ is lipchitz continuous, can I extend it to $\bar{A}$ and maintain the lipchitz continuity? [on hold]

Let $(X,d)$ be a complete metric space, $f:A\subset X \to X$ be Lipchitz continuous, does there exist an extension $\bar{f}:\bar{A} \to X$ such that $\bar{f}$ is also Lipchitz continuous?
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Limit and continuity

Hello I am a bit confused about this problem. It says, define a function f over the whole plane as $$f(x,y)=0$$ if $x=0$ and $$f(x,y)=0$$ if $y=0$ other wise defined by ...
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Show that $f(x)=0$ for all $x \in [a,b]$.

I have the following problem: Suppose that $f$ is continuous on $[a,b]$ and suppose that for all $x \in [a,b]$, $f(x) \geq 0$ and $f(x)\leq \int_a^x f(t)dt$. Show that $f(x)=0$ for all $x \in ...
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11 views

Global extremes and continuity of multivariable function

I am trying to find extremes and continuity of the function $$ g(x,y) = \frac{x}{y}. $$ I have found out that domain is simply $x \in \mathbb{R}$, $y \neq 0$. The derivative should be: $$ ...
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Investigate continuity of $\frac{x-y}{x^3-y}$

Investigate continuity of $$f(x) = \begin{cases} \frac{x-y}{x^3-y}, y\neq x^3\\ 1, y=x^3&\end{cases}$$ How to investigate that? Is it enough to show that when $y=x^3$ then the denominator is zero ...
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Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?

I know that there is a topological space $X$ which is not compact but such that , for any Hausdorff topological space $Y$ , any continuous function $f:X \to Y$ carries closed sets to closed sets . I ...
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22 views

For what values of $z$ is the following function differentiable? [closed]

For what values of z is $f(3)=|z|^2$ is differentiable?
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24 views

Proving that normalising a vector in $\mathbb{R^n}$ is continuous

Let $f:\mathbb{R^n}\backslash\{0\} \rightarrow \mathbb{R^n}\backslash\{0\}$ be the map $f(x) = \frac{x}{\lvert x\rvert}$. Here, we are using the standard topology on $\mathbb{R^n}\backslash\{0\}$, and ...
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Real Analysis continuous function question

Can anyone help me with the following question: Let $U = \lbrace x \in R: x>a \rbrace$, for some positive real number a, and let $f$ be a real-valued function on $U$. Define $lim_{x \rightarrow ...
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Show that the function kf defined by kf(x) = kx for all x∈[a,b] also satisfies the conclusion of the intermediate value theorem.

Let [a,b] be a closed and bounded interval, let k∈ℝ, and let f:[a,b]→ℝ be a function. Suppose that f satisfies the conclusion of the intermediate value theorem. Show that the function kf defined ...
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How do I prove that this function $f: \mathbb{R}^2 \to \mathbb{R}$ is not continuous at the origin?

My textbook gives the function $$ \begin{cases} \frac{x^2y}{x^4 + y^2} & (x, y) \neq (0, 0) \\ 0 & (x, y) = (0, 0) \end{cases} $$ as an example of a function that isn't continuous at the ...
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Let [a,b]$\subseteq \Bbb R$ be a closed bounded interval

Let [a,b]$\subseteq \Bbb R$ be a closed bounded interval, and let $f:[a,b] \rightarrow [a,b]$ be a function. Suppose that f is continuous. Prove that there is some $c \in [a,b]$ such that $f(c)=c$. ...
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1answer
26 views

Let $I, J\subseteq \Bbb R$ be open intervals, and let $f:I\to\Bbb R$ be a function.

Let $I, J\subseteq \Bbb R$ be open intervals, and let $f:I \to\Bbb R$ be a function. Suppose that $f$ is continuous. Let $x \in f^{-1} (J)$. Prove that there is an open interval ...
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Prove the function to not be continuous at $z = 0$

$$f(3) = \begin{cases} \dfrac{\mathrm{Re}(z)}{|z|} & \text{when $z \neq 0$} \\ 0 & \text{when $z = 0$} \end{cases}$$ Can someone please explain the concept behind solving such a problem? ...
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107 views

Where surjectivity goes in?

Let $X$ be an infinite set with the cofinite topology, and $f: X \to X$ a surjective function. Prove that $f$ is continuous if and only if $f^{-1}(\{x\})$ is finite for all $x\in X$. I know that $f$ ...
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Confusion about Lusin's Theorem.

I saw a proof which heavily relied on Lusin's Theorem recently, and I was hoping someone might be able to help me fill in the detail as to why this theorem allows for a particular creation. ...
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Finding a denumerable set $X_0$ satisfying a condition.

Let $(X,\tau)$, with $X$ an uncountable set, $x_0 \in X$ fixed, be the space with topology generated by the collection: $$\mathscr{B} = \{ \{x\} \mid x \in X \setminus \{x_0\}\} \cup \{ A \subset X ...
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If $g$ is continuous then $x^ng(x)$ converges on $[0,1)$

Suppose $g:[0,1]\to\mathbb R$ is a continuous function satisfying $g(1)=0$. Prove that the functions $f_n(x)=x^ng(x)$ converge uniformly on $[0,1]$. Hence or using Mean Value Theorem, prove that if ...
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53 views

Is $\{(x,y) \in \mathbb R^2 : xy=0 \}$ homeomorphic to $\mathbb R$?

Is $\{(x,0) : x \in \mathbb R \} \cup \{(0,y) : y \in \mathbb R \}$ homeomorphic to $\mathbb R$ ? I am totally stuck and I don't even have any intuition whether they should be homeomorphic or not . ...
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To find the value of a function at a point where it is continuous

Find $f(0)$ so that the function $f(x)=\dfrac{1-\cos(1-\cos x)}{x^4}$ is continuous everywhere. My attempt: By applying sandwich theorem $-1 \le cos(x) \le 1$. $$1 \ge -cos(x) \ge -1$$ $$2 \ge ...
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Families of continuous functions with non-continuous derivatives

What families of functions have the property of being continuous yet having a non-continuous derivative? And how many of these families are there? $$f(x) = \sqrt[n]{x}$$ when "n" is an odd number ...
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Finding the point at which a function is continuous

I am trying to understand the solution to the following question. At which $c\in\mathbb{R}$ is the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by $$f(x)=\begin{cases}x&\text{if $x$ is ...
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1answer
29 views

Show a series of functions is discontinuous at a point

I have a series of functions which converges to an integrable function. I need to show that this function is discontinuous at every point . For starters (because of the way it's defined) I'm just ...
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68 views

Monotone increasing continuous function with $\int_a^b f' = f(b) - f(a)$ which is not absolutely continuous

If $f:[a, b] \to \mathbb{R}$ is continuous and real-valued, f' integrable on [a, b], and $\int_a^b f' = f(b) - f(a)$, must f be absolutely continuous? What if f is monotone increasing? For the ...
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37 views

Define parameter so that function is continuous [closed]

So we have this function: $f(x)= \frac{24}{(3+x^2)}-\frac{a}{x-1}$ I need to find parameter "a" in order to make this function continuous...any ideas?
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Relation between Lipschitz condition and linear growth condition

If for a function $f:\mathbb{R}\rightarrow\mathbb{R}$ it is given that it satisfies a Lipschitz condition $\big|f(x)-f(y)\big| \le L\big|x-y\big|$, for all $x,y\in\mathbb{R}$, can we say anything ...
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37 views

Absolutely continuous iff continuous of bounded variation

I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not ...