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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If $f(x)$ and $g(x)$ are NOT both continuous at some point $c$, does this imply that $f(x) + g(x)$ is also not continuous at point $c$?

If $f(x)$ and $g(x)$ are NOT both continuous at some point $c$, does this imply that $f(x) + g(x)$ is also not continuous at point $c$? I know that if $f(x)$ and $g(x)$ are continuous then we know ...
3
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1answer
49 views

I've got a definition, but says something strange, what does it mean?

$M_1:=(M,d_1), \ \ M_2:=(M,d_2)$. $d_1$ is equivalent to $d_2$ if the identity $x\rightarrow x$ of $M_1$ over $M_2$ is an homeomorphism I'm not sure what it is talkin about when it says "identity" ...
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0answers
21 views

Looking for a (counter-example) two-variable function

Context : I've encountered two theorems about the derivation of parametric integral except one of the two needs the function to verify one more condition. I've tried to show that the condition is not ...
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1answer
30 views

what should be value of $f(x)$ to make $f(x)$ continuous at $x = 0$ where $f(x) = (\cos x)^{(1/x)}$

This was a problem asked to me in an interview. i am not sure about the continuity as on one side of 0 the function reaches 0 and on the other it tends to infinity. so can it be continuous?
2
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1answer
40 views

Continuous maps and compact-open topology

Let $X,Y,Z$ be three topology spaces. And let $f$ be a map from $X\times Y$ to Z. I want to show that, $f$ is continuous if and only if $f(\cdot,)$ (as a map from $X$ to $C(Y,Z)$, which equipped with ...
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3answers
38 views

Difference between a continuous function and an isometry? Is a continuous function a homomorphism?

Definition of continuous function on a set: Suppose $X$ and $Y$ are metric spaces. Let $f: X \to Y$. We say $f$ is continuous on $X$ if for every $\varepsilon >0$, $\exists \delta >0$ such that ...
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1answer
24 views

Limit of the composition of two functions with f not necessarily being continuous.

Let $g$ be a continuous function at a point $a \in \mathbb{R}$, and let $$\lim_{x\rightarrow g(a)} f(x) = L$$ Show that $$\lim_{x\rightarrow a} (f\circ g)(x) = \lim_{x\rightarrow g(a)} f(x)$$. ...
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1answer
24 views

If $f$ is continuous at $a$, then there is a number $\delta > 0$ such that f is bounded above and below on the interval $(a - \delta, a + \delta)$.

If $f$ is continuous at $a$, then there is a number $\delta > 0$ such that f is bounded above and below on the interval $(a - \delta, a + \delta)$. Since $f$ is continuous at $a$, $\exists \delta ...
2
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2answers
134 views

Is it possible for a function to be continuous at a point if the function is not defined either to the left or right of that point?

Take for example the function $f:[0,\infty) \to \Bbb{R}$ given by $f(x)=\sqrt x$ is this function continuous at $x=0$? For it to be continuous we look for $$\lim _{x\to0} f(x)$$ and we say it is ...
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2answers
45 views

Number of continuous functions, analysis question.

How many different continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ exist for which $$ (f(x))^2 = x^2,\qquad x\in\mathbb{R}? $$ I'm pretty sure there are only 4 $f(x) = x, -x,|x|,$ and ...
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1answer
38 views

Continuity and discontinuity of a function [closed]

How do we know from looking at the function, if it is continuous or discontinuous and at what points? How can a function be continuous if there are "gaps"? If you can, can you give the answer in as ...
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1answer
38 views

Showing $\chi_E$ restricted to $F$ is not continuous under certain conditions.

Here is a problem from a practice qualifying exam that I am having some trouble with. Let $E$ be a closed subset of $[0,1]$ with positive measure and dense complement. Show that if $F \subseteq ...
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1answer
33 views

Taking the limit of $r \to 0$ in $f(t) \leq Cg(r)$ when this inequality holds for all $ r > 0$

Good morning I have proved the following inequalty: $$f(t) \leq Cg(r) \quad \text{for all $t\in [0,T]$ and $r > 0$}$$ for two functions $f$ and $g$, both of which are continuous. Am I allowed to ...
0
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1answer
48 views

Question about differentiable function

Let $f:[0,1]\to R$ be a real-valued function which is continuous on $[0, 1]$, differentiable on $(0,1)$ and $f(0)=f(1)=0$, $f(1/2)=1$. prove the following statements $a$. There exists $\epsilon \in ...
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1answer
61 views

Prove $f$ is continuous on R

Prove that if $f$ is defined on R and continuous at $x_0=0$ and if $f(x_1+x_2) = f(x_1)+f(x_2)$ ∀ $x_1,x_2$ ∈ R , Then $f$ is continuous on R From $x_0$ being continuous we know $\displaystyle ...
0
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3answers
61 views

if two functions are not equal the how come limit of those two are equal?

Suppose we are having two functions $$f(x )=\frac{\sin x(4-x^2)}{4x-x^3}$$ and $$g(x)=\frac{\sin x}{x}$$ they are not equal everywhere(for all real numbers), but they both are equal only in ...
0
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1answer
19 views

Prove that the inclusion $i : A \rightarrow X$ is a continuous function, provided that $A$ has the subspace topology

Prove that the inclusion $i : A \rightarrow X$ is a continuous function, provided that $A$ has the subspace topology i.e. $A$ has the subspace topology $\implies$ $i : A \rightarrow X$ is a ...
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1answer
15 views

$X,Y$ euclidean spaces $(dimX=dimY)$; $f:U\subset X \to Y ; f \in \mathbb C^1$ If $f'(a):X \to Y$ is surjection then it is an isomorphism?..

$X,Y$ euclidean spaces $(dimX=dimY)$; $f:U\subset X \to Y ; f \in \mathbb C^1$ If $f'(a):X \to Y$ is surjection then the mapping of every ball with a center in $a$ includes a ball centered in $f(a)$. ...
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1answer
40 views

Number of values of $x\in [0,\pi]$ where $f(x)=\lfloor 4\sin x-7\rfloor$ is non derivable is?

Number of values of $x\in [0,\pi]$ where $f(x)=\lfloor 4\sin x-7\rfloor$ is non derivable is? $f(x)=\lfloor 4\sin x-7\rfloor=\lfloor 4\sin x\rfloor-7$ I drew the graph of the $f(x)$ and see that ...
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1answer
15 views

Continuity and differentiability problem and checking is the function elelment of $C^0$ and $C^1$

Find value of a for which function $f(x) \in C^0(R)$ and $f(x) \in C^1(R)$ f(x) = \begin{cases} |x|^a sin(\frac{1}{x}) & \text{if }x \neq0,\\ 0 & \text{if } x=0 \end{cases} I try this: I ...
0
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1answer
52 views

Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.

Let $f:\mathbb{R} \to \mathbb{R}$ be a function such that $|f(x)| \leq x^2$ . Prove whether or not the function is continuous and differentiable at $x=0$. Please tell me where am i wrong i have used ...
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1answer
27 views

Why isn't there equality in the definition of (upper) semicontinuity?

The "standard" definition of upper semicontinuity at a point $x_0$ in a metric space seems to be $\limsup_{x\to x_{0}} f(x)\le f(x_0)$. However, why is it this weaker condition instead of ...
2
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1answer
45 views

Does every right-continuous function have left limits?

Let $t \mapsto f(t)$ be a right-continuous function such that every $s\leq \infty$, $\lim_{q\uparrow s}f(q) $ exists and is finite. Here $q\uparrow s$ means we approach $s$ along the rationals. Is it ...
0
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1answer
39 views

Example of an open set that is discontinuous

Can anyone give me an example of a function $f: \Bbb R \rightarrow \Bbb R$ such that for any open subset $V$ in $\Bbb R$, $f(V)$ is open but $f$ is not continuous at any point. This was a side ...
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620 views

Example of a continuous function with a discontinuous inverse

What is an example of a function $f: \Bbb R^n \rightarrow \Bbb R^m$ such that $f$ is continuous and injective but that $f^{-1}$ is not continuous. Our professor teased us with the notion but I ...
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2answers
29 views

If $f$ is continuous on $[a,b]$, then $f$ is bounded on $[a,b]$. Have questions about the proof

If $f$ is continuous on $[a,b]$, then $f$ is bounded on $[a,b]$. First off, I have seen a proof of this and I sort of get it but wanted to go through the proof and justify my arguments correctly to ...
2
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0answers
35 views

$g(f(x))=f'(x)$ and $f$ is differentiable. Prove that $f$ is monotonic. [duplicate]

We have $f,g: \mathbb{R} \rightarrow \mathbb{R}$ and $g(f(x))=f'(x)$. $f$ is differentiable in all points. Prove that $f$ is monotonic. It is easy to prove that $f$ is monotonic when $f'$ is ...
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1answer
29 views

Finding an appropriate continuous functions

Q: Find examples of: 1) A continuous function $f : (0,1) \to \mathbb{R}$ which is bounded but does not attain a maximum or minimum on $(0,1)$ 2) A continuous function $f : \mathbb{R} \to ...
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1answer
19 views

Continuous image of dense set is also dense.

Let $f : X \to Y$ is continuous and $E \subset X$ is dense set. Then I want to show that $f(E)$ is also dense in $f(X)$ My approach is like this : Let $y \in f(X)\setminus f(E)$, then $\exists x \in ...
2
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2answers
36 views

Test whether $f$ is constant or not.

Let , $f:[0,1]\to \mathbb R$ be continuous and $f(x^2)=f(x)$ for all $x\in [0,1]$. Then which are TRUE ? (A) $f$ is a constant function. (B) $f$ is differentiable function. (C) $f$ is uniformly ...
28
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2answers
944 views

Do all continuous real-valued functions determine the topology?

Let $X$ be a topology space. If I know all the continuous functions from X to $\mathbb R$, will the topology on $X$ be determined? I know the $\mathbb R$ here is, somewhat, artificial. So if this is ...
3
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0answers
19 views

Is it okay if I considered $f$ as uniformly continuous and bounded on specific interval if $f$ is continuous?

Problem Suppose $f$ is continuous and $\phi$ is of bounded variation on $[a, b]$. Show that the function $\psi(x)=\int_a^xfd\phi$ is of bounded variation on $[a, b]$. If $g$ is continous on $[a, ...
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2answers
37 views

Does the limit exist?

I had a question about the definition of a limit. I know for a limit to exist the right hand limit must equal the left hand limit but what if the graph of a specific function has the domain from [0,5] ...
2
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1answer
20 views

Continuity and Box topology in $\mathbb{R}^\omega$

I am trying to understand discontinuity of the the following function: $$ h:\mathbb{R}\to\mathbb{R}^\omega \quad h(x):=\left(x,\frac{x}{2},\frac{x}{3},\ldots\right) $$ where $\mathbb{R}$ is given ...
2
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1answer
35 views

Prove existence of a certain point

Let $p(x)$ be an odd degree polynomial in one variable with co-efficients from the set of real numbers.Let $g:\Bbb R\to \Bbb R$ be a bounded continuous function.Prove that there exists an $x_0$ ...
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1answer
17 views

Random variables are determined by their characteristic functions proof

From the line $F_Y(t) = \lim_{n\to\infty} F_y(t_n)$, here the author is assuming that $t$ is not a continuity point of $F_Y$, and is being approximated by continuity points $t_n$, and the result ...
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0answers
8 views

What is “the set in which this* satisfies lipschitz condition”?

I'm very confused, I believe my teacher was not clear enough, maybe im missing something. $$1)y''+2ay'+by=0$$ In my "definition of lipschitz condition": G is the domain of a function f in which you ...
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0answers
18 views

Multivariable function, semi-continuity

Here is the task: Let $X$ be a metric space. Let $F$ be a multivariable function, namely $F: X \Rightarrow \Bbb{R^2}, F(x) = \{{(y_1,y_2)\in\Bbb{R^2}\mid y_1y_2=f(x)\}}$, where $f(x): X ...
0
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1answer
44 views

Continuity of a Function $f$

I've been studying different types of functions and I came across one on What is an example that a function is differentiable but derivative is not Riemann integrable, but I can't figure out why ...
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2answers
39 views

how do prove a function does not satisfy Lipschitz condition on a square?

I've got to prove $$f(x,y):=y^{2/3}$$ doesn't satisfy Lipschitz condition in $$G:=\{(x,y): 0\leq x\leq 1,-1\leq y \leq 2\}$$. But I have problems with denying the definition (not sure if that's the ...
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4answers
70 views

if $|f(k) | \le k$ for all integers $k$, does that mean $ |f(x)| \le |x|$ for all $x$ in $\mathbb R$?

if $|f(k) | \le k$ for all integers $k$, does that mean $ |f(x)| \le |x|$ for all $x$ in $\mathbb R$? Note that $f$ is uniformly continuous. This question is a follow up to a previous answer: ...
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1answer
17 views

Prove that Log is defined on D [closed]

$D=D(0,R)$ is the disk of center $0$ and radius $R$. Given that $a>R$ and $\Phi(z)=\frac{a-z}{a+z}$, I have proved that $\forall z\in D$, $\operatorname*{Re}(\Phi(z))>0$. Prove that $f = ...
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1answer
42 views

good source for learning continuity,discontinuity

What is a good online source, sites for learning continuity (specifically) . i have learnt limits and differentiation problem lies within the continuity. Any good websites(only) for learning this . ...
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1answer
49 views

$X,Y$ metric spaces , $X$ complete , $Z$ is Hausdorff , $f,g:X \times Y \to Z$ continuous in each variable and coincide on a dense subset , is $f=g$?

Let $X,Y$ be metric spaces , $X$ be complete and $Z$ be a Hausdorff space ; let $f,g:X \times Y \to Z$ be functions such that each of $f$ and $g$ is continuous in $x \in X $ for each fixed $y \in Y$ ...
0
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1answer
22 views

Use $\epsilon, \delta$ to show the function is continuous at $x_0=0$

Q: Show function is continuous at $x_0=0$ $$f(x) = \begin{cases}x\sin\frac{1}x & x > 0 \\ x^{1/3} & x \le 0 \end{cases}$$ Attempt: for any $\epsilon >0$ there exists $\delta >0$ ...
0
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1answer
47 views

Prove or disprove that $ f(x) < q|x|+p$ if and only if $f(x)$ is continuous

I need to prove that if $f$ is uniformly continuous then there exist $q,p > 0$ s.t $f(x) \le q|x|+p$. Then I need to prove or disprove that if $f(x) \le q|x|+p$ then $f$ is uniformly continuous. I ...
2
votes
1answer
72 views

Show that $f$ is continuous.

Let $f:\Bbb R\to \Bbb R$ be a function which maps intervals to intervals .Suppose for each sequence $x_n$ converging to $x$ there exists a constant $M$ such that $$\lvert f(x_n)-f(x)\rvert\le ...
1
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3answers
46 views

Intermediate-Value Theorem - Find roots of an equation

I've an homework question where i need to prove that the following equation contains at-least three roots $ {x^4 \over 10} = {x^4-100 \over x-1} $. I was able to find three roots after redefining ...
0
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2answers
35 views

Prove that if $f$ and $g$ are uniformly continuous on A and are both bounded on A, then $fg$ is uniformly continuous on A.

Let $f$ and $g$ be uniformly continuous on A. Then given $\epsilon >0$ there exists a $\delta_{1} > 0$ such that if $|x-y| < \delta_{1}, \forall x,y \in A$, then $|f(x)-f(y)| < ...
1
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0answers
38 views

The action of a topological group on the function space is continuous?

Sorry for my bad english. Let $X$ and $Y$ be two topological spaces, and $G$ a topological group, let $\theta : G \times X \to X$ be a continuous action of $G$ on $X$. We defined the action of $G$ on ...