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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) ...

3
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1answer
44 views

let $f,g:\mathbb{I} \to \mathbb{I} $ continuous functions

$\mathbb{I} = [0,1]$ let $f,g:\mathbb{I} \to \mathbb{I} $ continuous functions such that $ f \circ g = g \circ f$. Prove that there is $ x_0 \in \mathbb{I}$ such that $ f(x_0) =g(x_0)$. Could you ...
0
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0answers
9 views

How to prove a function is jointly continuous

$F(x, t) = \sin(tx)/x$. Why is $F(x,t)$ is jointly continuous on $x\in \mathbb{R}\setminus\{0\}$ and $t\in\mathbb{R}$. I tried to use the fact that $\sin(x)/x$ is continuous for $x\in\mathbb{R}\...
3
votes
1answer
34 views

Give an example of a real valued function on $[0,1]$ satisfying the following property.

My whole question looks like- Give an example of a real valued function $f:[0,1]\to\Bbb{R}$ which satisfies the following property- For each $y\in\Bbb{R}$ either there is no $x\in[0,1]$ such that $f(x)...
0
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0answers
11 views

$f:[0,1]\to\Bbb{R}$ has property- $\forall y\in\Bbb{R}$, either $\nexists x\in[0,1]$ s.t. $f(x)=y$ or $\exists$ exactly two such points in $[0,1]$.

My whole question looks like- A real valued function $f:[0,1]\to\Bbb{R}$ has the property that $\forall y\in\Bbb{R}$, either $\nexists x\in[0,1]$ s.t. $f(x)=y$ or $\exists$ exactly two such ...
1
vote
1answer
27 views

If $f$ is continuous, then $\exists \delta > 0$ so that $\vert f(x) \vert \geq \frac{\vert f(x_0) \vert}{2} > 0$

Problem: Let $\, f : \mathbb{R} \rightarrow \mathbb{R}$ be continuous at the point $x_0$ and $f(x_0) \ne 0 \ \forall x \in \mathbb{R}$. Show that $\exists \delta > 0$ such that $\vert f(x) \vert ...
2
votes
1answer
75 views

Prove that $f(x)$ is continuous over $\mathbb{R}.$

Problem Let $f(x)$ be continuous at $x=0$ and satisfy that $f(0)=0$, $$3f(x)-4f(4x)+f(16x)=3x ,\forall x \in \mathbb{R}.$$ Prove that $f(x)$ is continuous over $\mathbb{R}.$ Comment Maybe, we can ...
1
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1answer
52 views

Where is $\sqrt{z+1}$ analytic and continuous?

I am trying to determine where $$f(z)=\sqrt{z+1}$$ is analytic, where the square root is the principal branch. I know that $\sqrt{w}$ is analytic for $\mathbb{C}\setminus(-\infty,0]$. So, I think $f(...
2
votes
1answer
49 views

Proof that if f(a)<0 and f(b)>0 f and is continuous on [a,b] then f changes sign at some c in (a,b)

In Calculus, people argue that if $f(a)<0$ and $f(b)>0$, then IVT guarantees that $f(x)$ changes sign at some point in $(a,b)$. I understand that IVT guarentees that there is a $c$ such that $f(...
2
votes
1answer
33 views

Is a continuous function with derivative zero a.e. of bounded variation?

Let $f:[a,b] \to \mathbb{R}$ be a continuous function and suppose that $f$ is differentiable a.e. and that $f'(t) = 0$ for almost all $t \in [a,b]$. Is it true, that then $V_a^b(f) < \infty$, where ...
0
votes
2answers
33 views

Limit as $z$ approaches $0$ for $e^{1/z^4}$ [duplicate]

I want to work out if the limit as $z$ approaches $0$ for $e^{1/z^4}$ exists and if not then why. I worked out that the left and right sided limits both equal to +∞ so I thought that was enough to ...
0
votes
1answer
23 views

Continuity of a function and bound

Please, check my solution. Task: let $f(x): \mathbb{R} \rightarrow \mathbb{R}$. If $f(x)\le M$ for every $x\in\mathbb{Q}$ then is is also $\le M$ for every irrational number. Me solution: if it is ...
0
votes
0answers
25 views

Prove the continuity

$f(x): [0, 1) \rightarrow \mathbb{R}$. Prove that the function $f(x) = \sum_{n=1}^{\infty}2^{-n}\{\frac{[2^nx]}{2}\}$ is continuous. Please, give me a hint where to start (I want to prove it using ...
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1answer
49 views

Prove the continuity $f(x) = \sum_{n=1}^{\infty} \sin(\frac{x}{n!})$ [on hold]

Prove that $\displaystyle f(x) = \sum_{n=1}^{\infty} \sin\left(\frac{x}{n!}\right)$ is continuous for all $x\in\mathbb{R}$. I am trying to prove it with $\varepsilon$ and $\delta$, but did not ...
0
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0answers
39 views

Uniform continuity on positive reals!! [on hold]

for what values of $a,b>0$ $$x^a\cdot \sin(x^b)$$ is uniformly continuous on positive reals? Atleast I have concluded that $a$ should be less than $1$. Can anyone help me to proceed further?
0
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0answers
24 views

Give an example of a set $S$ whose support function is not continuous on $R^n$.

Let $A \subseteq \mathbb{R}^n$. The support function of set $A$ is defined as the following $$ S_A(x)=\sup_{y \in A} x^Ty $$ where $x \in \mathbb{R}^n$. Notice that when $A$ is bounded the function ...
-3
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0answers
30 views

Is the function $\sin(x) + 1$ continuous? [on hold]

Consider $f : [0,1] \to \Bbb{R}$ by $$f(x) = \sin(x) + 1$$ Is $f$ continuous?
-1
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0answers
39 views

Provide a counterexample that shows supremum over an unbounded set is not continuous.

Let $A \subseteq \mathbb{R}^n$ and $S_A(x)$ be defined as the following $$ S_A(x)=\sup_{y \in A} x^Ty $$ where $x \in \mathbb{R}^n$. Give an example of a set $A$ for which $S_A(x)$ is not continuous ...
0
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2answers
32 views

Continuity and its definition

Is it true that a function $f$ is continuous at point $a$ if the following holds? $$ \exists\varepsilon > 0 \ \ \ \exists\delta>0 \text{ such that }\ \ |x - a|<\delta \Rightarrow |\frac{f(...
0
votes
2answers
22 views

Proving a removable discontinuity exists

So I made up the following function as a thought exercise (for myself). $$f(x) = 1\hspace{0.3cm} \forall\hspace{0.1cm} x = 1,2,3 \hspace{0.4cm}\text{and} \hspace{0.4cm} -1 \hspace{0.3cm}\forall \...
0
votes
0answers
22 views

What happens to continuous functions when you add metrics?

Let $g(x) = ax$ be multiplication by $a$ where $g : \Bbb{N} \to \Bbb{N}$ and we're given the topology induced by the pseudometric $d(x,y) = |\psi(x) - \psi(y)|$ where $\psi(xy) = \psi(x) + \psi(y)$ ...
0
votes
1answer
39 views

Does $f$ absolutely continuous imply $|f|$ absolutely continuous?

$f: [a,b] \to \mathbb{R}$ is absolutely continuous $\Leftrightarrow$ $$\forall \epsilon>0 \ \exists \delta>0: \sum_{i=1}^{n}(b_i-a_i)<\delta \Rightarrow \sum_{i=1}^{n}|f(b_i)-f(a_i)|<\...
1
vote
1answer
22 views

Do continuous functions satisfy the “right-above right-below limit property?”

Let $\Omega = (a,b) \subseteq \mathbb{R}$, $f: \Omega \to \mathbb{R}$. We say that $f$ does NOT satisfy the "right-above right-below limit property" (my own naming for this) if and only if $\forall \...
1
vote
1answer
42 views

How to show the continuity of a specific function?

Let $X$ be a closed interval of $\mathbb{R}$, and $C(X)$ be the Banach space of all real-valued continuous functions defined on $X$. Denote by $C(X)_+$ the set of all non-negative functions in $C(X)$. ...
2
votes
1answer
39 views

Show function is continuous given continuity of another function $\mathbb{R}^n \rightarrow{}\mathbb{R}$

Given that $f: \mathbb{R}^n \rightarrow{} \mathbb{R},\ f(\mathbf{x}) = \max\{|x_1|,...,|x_n|\}$ is continuous, show that $g: \{\mathbf{x} = (x_1, x_2) \in \mathbb{R}^2 | x_1, x_2 \geq 1\} \rightarrow{}...
0
votes
1answer
15 views

Uniform continuity and bounded sequence

Problem If the sequence ${f_n}\subset C[a,b]$ is uniformly continuous, it is bounded. If the sequence ${f_n}\subset C[a,b]$ is bounded, it is uniformly continuous. Maybe someone can give me some ...
1
vote
2answers
51 views

Show that supremum of over a bounded set is continuous.

Let $A \in \mathbb{R}^n$ $S_A(x)$ is defined as the following $$ S_A(x)=\sup_{y \in A} x^Ty $$ where $x \in \mathbb{R}^n$. Show that if $A$ is a bounded set, then $S_A(x)$ is a continuous function. ...
1
vote
1answer
41 views

Spivak, Continuity (An exercise)

I'm working on Spivak and I've come across this problem. Suppose that $f$ satisfies $f(x+y) = f(x) + f(y)$, and $f$ is continuous at $0$. Prove that $f$ is continuous at a for all a. OK, then the ...
0
votes
2answers
45 views

Show that if $\pi_i \circ f$ is continuous for each i where $f$ is a function $f: \mathbb{R}^n \rightarrow{} \mathbb{R}^m$, $f$ is continuous

Show that if $\pi_i \circ f$ is continuous for each i where $f$ is a function $f: \mathbb{R}^n \rightarrow{} \mathbb{R}^m$$\iff$ $f$ is continuous. Where $\pi_1, ..., \pi_m: \mathbb{R}^m \rightarrow \...
1
vote
1answer
31 views

How to prove that this function is continuous but not uniformly continuous?

I am having some troubles solving a question on my homework sheet: Prove that $\,\,f:\Bbb{R}^n\setminus\{0\}\to \Bbb{R}^n$, defined as $$f(x) = \frac{x}{||x||}$$ is continuous but not uniformly ...
0
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0answers
29 views

Dense set in the set of continuous functions…

Consider the set $C(\mathbb R, \mathbb R)$ of all continuous functions from $\mathbb R$ to $\mathbb R$. Is it possible to construct a dense countable set of $C(\mathbb R, \mathbb R)$ consisting of ...
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votes
2answers
29 views

Why does a uniformly continuous function on [a,b] in the reals need be closed AND bounded?

I cannot come up with any counter examples as to why a function would need to be both closed and bounded to be uniformly continuous. Why is it not sufficient to just have one condition? For example, ...
0
votes
2answers
32 views

Use the definition of limit to show that $\lim_{(x,y) \to (5,4)} 2x^2 -3y^2 = 2$.

For any $\epsilon > 0$, I want to show that $|2x^2 -3y^2-2|< \epsilon$ whenever $0<\sqrt{(x-5)^2 + (y-4)^2}< \delta$. What I've tried so far is expanding out the argument in the ...
4
votes
5answers
40 views

If limit of $f$ is $L$ and limit of $g$ is $M$, then limit of $g$ composed $f$ is $M$?

Problem: Find examples of functions $f$ and $g$ defined on $\mathbb{R}$ with $\lim\limits_{x\to a}f(x) = L$, $\lim\limits_{y\to L}g(y) = M$, and $\lim\limits_{x\to a} g(f(x))\neq M$. I have tried ...
4
votes
2answers
70 views

Proving continuity in the origin $\frac{xy}{\sqrt{x^2 + y^2}}$

Let $g: \mathbb{R^2} \to \mathbb{R}$. How can I prove that $g$ is continuous in its origin, but not totally differentiable? If I take $$g(\frac{1}{n},\frac{1}{n}) = \frac{1}{n\sqrt{2}} \to 0 \...
1
vote
0answers
22 views

Prove $C(t)$ varies in an absolutely continuous way

Prove that there exists an absolutely continuous function $v(\cdot): I → R$ such that for any $y\in\mathbb{R}^n$ and $s,t\in [0,T]$, $$|dist (y, C(s)) − dist (y, C(t))| ≤ |v(s) − v(t)|$$ for all $s,t\...
1
vote
0answers
28 views
+100

Proving $(x^4-y^4) \cos (\frac{1}{\left\lVert (x,y) \right\rVert^3_2})$ is totally differentiable

How can one prove, that this function is totally differentiable on $\mathbb{R^2}$ and not continuous partially differentiable on $\mathbb{R^2}$? I know that to prove the total derivative one first ...
1
vote
1answer
30 views

Show that the support function of a bounded set is continuous.

The support function of a set $A \in \mathbb{R}^n$ is defined as the following $$ S_A(x)=\sup_{y \in A} x^Ty $$ where $x \in \mathbb{R}^n$. Show that the support function of a bounded set is ...
1
vote
0answers
40 views

Problem regarding proving a real valued function to be continuous. [duplicate]

My question is - Let, $f:[a,b]\to\Bbb{R}$ be a continuous function. Define $g:[a,b]\to\Bbb{R}$ by $$ g(x)= \begin{cases} f(a), & \text{when}\quad x=a\\ \operatorname{sup}_{(a,x]} f, & \...
-3
votes
1answer
37 views

Absolute continuous functions property. Is it true? [on hold]

Let $f,g:[0,T]\to [0,\infty)$ be two absolute continuous functions such that: $f(0)=g(0)>0$. We know that there is a sequence $(x_n)_n$ converging to $0$ such that: $$f(x_n)\neq g(x_n)$$ Can we ...
0
votes
2answers
36 views

Show that $f$ is continuous at $\frac12$

Let $f$: $[0,1] \to [0,1]$ be defined by $$f(t):=\begin{cases} t& \text{if}\; x \in \Bbb{Q}\\1-t &\text{otherwise}\end{cases}$$ I have to show that $f$ is only continuous at $t=$$\...
1
vote
0answers
13 views

Continuous function, monotonic converging sequences

In our lecture notes our lecturer wrote "f is continuous iff f(xn) → f(x) for all xn→ x strictly monotone." I was wondering...the definition is fine even if I don't restrict myself to strictly ...
0
votes
0answers
27 views

Non-zero affine function vanishing on a compact convex proper subset

Let $K_1, K_2$ be two convex, compact spaces. Denote by $A(K_1), A(K_2)$ the continuous affine functions on $K_1$ and $K_2$, respectively. Let $\varphi: K_2\to K_1$ be a given continuous, affine ...
0
votes
3answers
58 views

What is a plain English explanation of “$\epsilon$-$\delta$ criterion” and how to use it for proving continuity?

I would prefer as little formal definition as possible and simple examples on how to proof that any given function is whether continous or not with the help of the criterion.
1
vote
1answer
13 views

Neither piecewise continuous function nor continuous function examples

After I read the definition of piecewise continuous function, it seems like either function is piecewise continuous or continuous. Can a function be neither of these two?
0
votes
0answers
19 views

Example of two non-homeomorphic spaces with two continual mappings between them

My question is about to find to topological spaces $(X,\tau_X),(Y,\tau_Y)$ that are non-homeomorphic but there exist two bijective functions $f:X\to Y, f \in C(X \to Y)$ and $g:Y \to X, g \in C(Y \to ...
0
votes
0answers
35 views
+50

Proving continuity of a multivariable function by “reducing” it to a single variable function

My goal is to examine the continuity of a certain multivariable function. In that regard, (mainly because my math in the specific area is dusty) I find it somewhat tricky to either prove or disprove ...
0
votes
1answer
20 views

Find a continuous bijection with an noncontinuous inverse in metric spaces

I am asked to find a continuous bijection $f$ from $[0,1)$ to $S$, when $S = ${$(x,y) | x^2 + y^2 \le 1$}, so the it inverse function would be noncontinuous. In other words, $S$ is the boundaries of ...
0
votes
1answer
25 views

Why isn't this a contradiction? (Preimage of an open set)

My question arises from the combination of both following theorems: Theorem 1: Every open set is a continuous image of a closed set. That is, for every open set $A\subseteq \mathbb{R}^m$ there is a ...
0
votes
1answer
36 views

Show that the image of a closed set is closed given that f is continuous and the preimage of a compact set is compact

Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous and has the additional property that $f^{-1}(K)$ is compact whenever $K$ is compact. Show that $f(C)$ is closed whenever $C$ is closed. I have ...
0
votes
1answer
39 views

Any function $f: \mathbb{Z} \to \mathbb{R^n}$ is continuous.

Definition of a limit: Let $S \subset \mathbb{R^n}$ and let $f$ be a function from $S$ tinto $\mathbb{R^m}$. If $a$ is a limit point of $S$ then a point $v \in \mathbb{R^m}$ is the limit of $f$ at $...