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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Let $f:[0,\frac{\pi}{2}]\to R$ be $f(x)=max\{x^2,cosx\}$.Prove $f(x)$ attains minimum at $x_0$ and is a sulution to $x^2=cosx$

I try to write $f(x)=\frac{1}{2}x^2+\frac{1}{2}cosx+\frac{1}{2}|x^2-cosx|$ and use the Extreme Value Theorem to show that $x_0$ exists in $[0,\frac{\pi}{2}]$, but I don't know how to show the seconde ...
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Boundedness Theorem for continuous functions on intervals

Just want to confirm this is a suitable proof: Assume $f$ is not bounded on $I$. So, for any $n \in \mathbb{N}$, $\lvert f(x)\rvert > n$. Since $I$ is bounded, $x_n$ is also bounded. By ...
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$f(x) =\ln(2x^2 + 1)$ is continuous on $\mathbb{R}$

True or False The function $f : \Bbb R \to \Bbb R$ defined by $f(x) = \ln(2x^2 + 1)$ is continuous on $\Bbb R$. I know this condition that The function $f$ is continuous at some point $c$ of its ...
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Let $f,g$ be continuous from $\mathbb R$ to $\mathbb R$ [duplicate]

Let $f, g$ be continuous from $\mathbb R$ to $\mathbb R$, and suppose that $f(r) = g(r)$ for all rational numbers $r$. Is it true that $f(x) = g(x)$ for all $x \in \mathbb R$?
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I need help finishing this proof using the Intermediate Value Theorem?

Let $f$ and $g$ be continuous functions on $[a,b]$ such that $f(a)\geq g(a)$ and $f(b) \leq g(b)$. Prove $f(x_0)=g(x_0)$ for at least one $x_0$ in $[a,b]$. Here's what I have so far: Let $h$ be a ...
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Test for uniform continuity

Test for uniform continuity the function $ f(x, y) = (x^2 + y^2)^\alpha \sin{\frac{1}{x^2+y^2}} $ in $ \{ x^2+y^2 > 1\} $ If we consider $ \alpha < 1 $, then $ \lim_{\sqrt{x^2+y^2} \to ...
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45 views

Expanding a function

Is it possible to expand a function $$ f(x,y) = \dfrac{\sin (xy)}{\sqrt{x^2 + y^2}} $$ so it will be continuous on $\mathbb{R}^2$? Now, the denominator should not be equal to $0$, so for the domain, ...
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1answer
16 views

Show that if $g((x_n)) \rightarrow l$ and $g((y_n)) \rightarrow m$, then $l=m$

Suppose that $g: (a,b] \rightarrow \mathbb{R}$ is uniformly continuous. Suppose that both $(x_n), (y_n)$ are sequence in $(a,b]$ which converge to $a$. Show that if $g(x_n)) \rightarrow l$ and ...
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Continuous multivariable function without limits in a point

I am curious, if there can be a function $f(x,y)$, which is continuous in a point $[0,0]$, but for which iterated limit $\lim _{x \to 0} \lim _{y \to 0} (f(x,y))$ does not exist. Is it even possible ...
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26 views

Show that $f$ is uniformly continuous.

Suppose that $F:(a,b] \rightarrow \mathbb{R}$ is continuous and that the limit as $x \rightarrow a$ of $f(x)$ exists. Show that $f$ is uniformly continuous. I am really struggling with this one. HELP ...
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22 views

Counter Example to Tietze Extension Property for Arbitrary Topological Space

Above is my question. My only issue is the final bit! For statements $1.$ and $2.$, the answer is true, since in both cases $Y$ is normal and we know that both metric and compact, Hausdorff spaces ...
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1answer
15 views

Need Help look at function continuity

Consider the following piece-wise function: $$f(x, y) = xy \frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(0,0)= 0$. Discuss the continuity of $f$ at $(0, 0)$. Calculate $\partial f / ...
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19 views

A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
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1answer
29 views

Proving the continuity of functions from metrics [on hold]

Context: I'm studying mathematics at university, and am having trouble with some of the continuity questions. The following is a question from a previous assignment that I was unable to complete. The ...
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24 views

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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62 views

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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1answer
29 views

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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18 views

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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43 views

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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25 views

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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40 views

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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25 views

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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35 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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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? [closed]

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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1answer
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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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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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1answer
51 views

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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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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20 views

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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1answer
70 views

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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2answers
31 views

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? ...