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) ...

learn more… | top users | synonyms (1)

0
votes
1answer
50 views

$f:[a,b]\rightarrow \mathbb{R}$ is continuous on a closed set $F$. Modification of $f$ to be continuous on $[a,b].$

Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous on a closed set $F\subset[a,b]$. We can assume that $[a,b]\cap F^{c}$ is a disjoint union $\cup^{\infty}_{j=1} (a_j,b_j)$. $$\phi(x) :=\begin{...
-3
votes
1answer
52 views

Prove that f is constant under those conditions [on hold]

Let $f:\mathbb{R}\to \mathbb{R}$ be continuous at $0$ and $1$ and assume further that $f$ satisfies the functional equation $$f(x^2)=f(x).$$ Prove that $f$ is constant.
0
votes
2answers
44 views

$X$ be a Hausdorff topological space , let $f:X \to X$ be a continuous map such that $f\circ f=f$ , then is $f(X)$ is closed in $X$ ?

Let $X$ be a Hausdorff topological space , let $f:X \to X$ be a continuous map such that $f\circ f=f$ , then is it true that $f(X)$ is closed in $X$ ?
2
votes
0answers
28 views

$A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?

Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?
0
votes
2answers
36 views

Show that $f(x) = x^{{1}/{3}}$ is Hölder on $\mathbb{R}$

I am trying to show that $f(x) = x^{{1}/{3}}$ is Hölder continous on $\mathbb{R}$, but I have not been able to make much progress. To show that this is indeed true we would need to show that $$\left|...
0
votes
1answer
23 views

$S$ be $\pi$-system on a set, given two measures on $\sigma(S)$, is there a topology on $\sigma(S)$ making $S$ dense, and the two measures continuous?

Let $\Omega$ be a non-empty set , $S \subseteq \mathcal P(\Omega)$ be a Pi system (https://en.wikipedia.org/wiki/Pi_system ) on $\Omega$ , let $\sigma(S)$ be the $\sigma$-algebra generated by $S$ (i.e....
0
votes
3answers
27 views

How to find E(x) and Var(x) in this specific continuous probability distribution.

I've got into some confusion on continuous probability distributions, and everything related to it. This is the problem: Problem Image. I assume from the sketch that pdf is $f(x) = x$ for values of $x$...
0
votes
0answers
18 views

Solutions to Strum-Louiville equation continuous even with discontinuous coefficients?

In the physics paper here (should be open access), the author first studies a Schrödinger equation in the form of a Strum-Louiville equation $$\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx}\phi(x) = -\...
1
vote
2answers
47 views

A proof of the Continuity of the inverse matrix function

I would like to see a proof to this fact. If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there ...
1
vote
1answer
51 views

Proving that $\sin(1/x)$ is not continuous at 0

Let $$f(x) = \begin{cases} 0 &\text{ if $x=0$}\\ \sin(1/x) &\text{ otherwise} \end{cases} $$ Prove that $f$ is discontinuous at $0$ My proof goes like this: for the function to be continuous ...
1
vote
1answer
29 views

Continuity Of Complex Valued Functions

Consider the following complex valued function: $$f(z)=(2+z)Arg(z)$$ Does $f(z)$ have removable discontinuities? (Note: $Arg(z)$ denotes the principal argument.) The following is my approach: We ...
1
vote
1answer
49 views

At which points is the following function continuous?

From Royden and Fitzpatrick's Real Analysis, Fourth Edition (Chapter 1, Problem 48): Let $f$ be the function defined by $$ f(x) = \begin{cases} x & \text{if $x$ is irrational} \\ p \cdot \sin \...
0
votes
0answers
60 views

Continuity of a function on a square

Fix some $\ell\in\mathbb{R}^+$, and say I have a function $f:[0,\ell]\times[0,\ell]\to\mathbb{R}$ with the following properties: $f(s,t)$ is continuous everywhere except when $s=t$, where it is ...
1
vote
1answer
29 views

How to show continuity of a function with $n-1$ exponentiations?

Say we are given a function $$\Gamma(x)=f_1(x)^{f_2(x)^{\cdot^{\cdot^{f_n(x)}}}}$$ where $f_i,i\in[1;n]$, are continuous functions in their domains. Also assume that the function makes sense, e.g., ...
0
votes
2answers
55 views

Continuous functions and metric topology [on hold]

Let $X = C[0, 1]$ be the set of all continuous functions $f : [0, 1] \to \Bbb{R}$ (where the domain and codomain have their usual topologies). Let $d_1 : X \times X \to \Bbb{R}$ be the metric on $X$ ...
1
vote
0answers
25 views

A problem about a continuous iterated function [duplicate]

Let $f:\mathbb {R} \rightarrow \mathbb { R } $ be a continuous function such that $f\circ f \circ f=\text{id}_\mathbb{R} $. Show that $f=\text{id}_\mathbb{R}$. Is there any hint to prove this? ...
1
vote
2answers
28 views

Limits, and Continuity - Finding whether a function is continuous or not

$$ f(x) = \lim_{n \to \infty} \frac{\log(2 + x) - x^{2n}\sin x}{1 + x^{2n}} $$ then, check the continuity of the function at $x = 1$. I found this question in a text. After some thinking I ...
1
vote
1answer
30 views

Determining if this mapping is continuous?

Let $X$ be a closed and bounded subset of $\mathbb{R}^p$ and let $C(X)$ denote the vector space of continuous functions from $X$ to $\mathbb{R}$. For $f,g \in C(X)$, let $$ d_{\infty} (f,g) = \sup \...
1
vote
2answers
44 views

Equivalent definitions of continuity at a point

I'm going with a definition of a map over defined on topological spaces $f:X\rightarrow Y$ is continuous at a point $x\in X$ is as follows: $f$ is continuous at each element $x\in X$ if and only ...
5
votes
2answers
61 views

$f: \mathbb{R} \to \mathbb{R}$ integrable, $F(x) = \int_a^x f(y)\,dy$, $F$ necessarily continuous

Suppose $f: \mathbb{R} \to \mathbb{R}$ is integrable, and we define$$F(x) = \int_a^x f(y)\,dy.$$Why does it follow that $F$ is necessarily a continuous function?
0
votes
0answers
27 views

Discuss the continuity and differentiablity of given function.

If $\big[\cdot\big] $ denotes floor function (i.e the integral part of $x$) and $$f(x)=\big[x \big] \left(\frac{\sin \frac{\pi}{\big[x+3\big]}+\sin \pi \big[x+3\big]}{3+\big[x \big]} \right)$$, then ...
2
votes
1answer
15 views

The true definition of invariant functions on Matrix algebra

According to terminologies in "Invariant theory" a true definition for an invariant function $f:M_{n}(\mathbb{R})\to \mathbb{R}$ is the following: Definition 1: A continuous function $f$ is ...
1
vote
2answers
38 views

Proof that the nowhere differentiable functions are dense in $C_b(\mathbb R)$.

I tried to make a proof, where I use a Weierstrass function. I was surprised at how easy it was, and thus a little doubtful as to the correctness of the proof. I've looked it over, and didn't find any ...
3
votes
1answer
98 views

$f:\mathbb R \to \mathbb R$ be continuously differentiable function such that $f(x),f'(x)>0$ for all real $x$ , then $\lim _{x \to -\infty}f'(x)=0$?

Let $f:\mathbb R \to \mathbb R$ be a continuously differentiable function such that $f(x)>0 , f'(x)>0 , \forall x \in \mathbb R$ , then is it true that $\lim _{x \to -\infty}f'(x)=0$ ? I can ...
-1
votes
2answers
45 views

Find the following one sided limits algebraically?

$$\lim_{x\to 1^+}\frac{\sqrt{2x}(x-1)}{|x-1|}$$ $$\lim_{x\to 1^-}\frac{\sqrt{2x}(x-1)}{|x-1|}$$ I know how to find one-sided limits graphically, but not algebraically. There's gonna be 1 answer for ...
1
vote
0answers
31 views

Continuity of an integral with a removably discontinuous integrand

Say I have a function $f_c:[0,\ell]\to\mathbb{R}$, where $c\in K\subseteq\mathbb{R}^n$, with $K$ compact. For every choice of $c$, $f_c$ is continuous everywhere except at some $s_c\in[0,\ell]$, ...
1
vote
3answers
49 views

Find the following one-sided limits?

$$\text{a)} \ \ \lim_{x\to-2^+}(x+3)\frac{|x+2|}{x+2}$$ $$\text{b)} \ \ \lim_{x\to-2^-}(x+3)\frac{|x+2|}{x+2}$$ The answers are: $$\text{a)} \ \ 1$$ $$\text{b)} -1$$ How do you find them? It is ...
4
votes
0answers
43 views

Baby Rudin Exercise 4.13 Alternate Proof Verification

I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function ...
-1
votes
2answers
69 views

Prove that a sum of continuous functions is continuous

$\forall n\in N$, let $f_n(x): [0,1]\rightarrow \mathbb{R}$ be continuous functions and $M$ a positive integer. If $\forall x \in [0,1]$, $\lvert\sum_{n=1}^{\infty} f_n(x)\rvert \lt M$, then $\sum_{...
15
votes
8answers
3k views

what is sine of a real number

I never understand what the trigonometric function sine is.. We had a table that has values of sine for different angles, we by hearted it and applied to some problems and there ends the matter. Till ...
1
vote
4answers
106 views

How do you find one-sided limits *algebraically*?

Find $$\lim_{x\to\ -0.5^-}\sqrt{\frac{x+2}{x+1}}$$ Sorry, I have no idea where to start. I know how to find regular limits algebraically, but not one-sided. Thanks
0
votes
1answer
28 views

If a function is continuous a.e., then it is measurable. [duplicate]

Is this true or wrong? How to prove it ?
0
votes
2answers
63 views

$X$ is metric space s.t. for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ ; is $X$ compact?

Let $X$ be a metric space such that for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ , then is $X$ compact ? Compare with this $A \subseteq \mathbb R^n $...
2
votes
1answer
48 views

$A \subseteq \mathbb R^n $ s.t. for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ , is $A$ closed $\mathbb R^n$?

Let $A \subseteq \mathbb R^n $ such that for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ ; then I know that $A$ is bounded ; my question is , is $A$ closed in $\...
0
votes
2answers
26 views

Sequence criterion for continuity

Let $f:X\rightarrow Y$ where $X$ and $Y$ are topological spaces. Is it true that $f$ is continuous at a point $c$ iff for every sequence $x_n$ converging to $c$, we have $f(x_n)$ converging to $f(c)$?
2
votes
0answers
66 views

Calculus, continuity of the derivative at point [closed]

Let $f:I \to \mathbb{R}$ differentiable in the interval $I$ and let $a\in I$. For any sequences $(x_n)_{n\in \mathbb{N}}$ and $(y_n)_{n\in \mathbb{N}}$ in $I$ such that $\lim x_n = \lim y_n = a$, $x_n ...
1
vote
2answers
43 views

Prove or Disprove that $f(x)$ cannot be $C_3$

Consider a function $y=f(x)$ with a single argument x with the real number line as its domain. Fix a real number $Q>0$, and suppose the following: $f(x)=f'(x)=f''(x)=0$ for all x from the interval ...
0
votes
1answer
35 views

Two sequences have the same limit

Let $f$ and $g$ be real-valued continuous functions on $\Bbb R^2$ that satisfy the following condition: $$ x<y \implies x< f(x,y) < g(x,y) <y $$ Assume that there are two sequences $\{a_n\...
1
vote
2answers
33 views

Conflict between homeomorphism definition and continuity of inverse theorem

In the course of trying to invert a particularly nasty hormeomorphism candidate, I decided to look for theorems that can tell me when an inverse is continuous without actually having to invert. I have ...
1
vote
1answer
47 views

Equivalence of pretopological continuity to customary continuity (a direct proof)

Can you provide a direct (not based on the neighborhood definition of pretopologies) proof for pretopological spaces (expressed as closure operators) that a function $f$ from a topological space $\mu$ ...
0
votes
1answer
48 views

Limits of the function given by the following graph

$$ f(x)=\left\{ \begin{array}{c} 3-x, \ \ \ x<2 \\ (x/2) +1, \ \ \ x>2 \\ \end{array} \right. $$ (a) Find $$\lim_{x\to 4^+}f(x) \text{ and} \lim_{x\to 4^-} f(x)$$ (b) Does $$\lim_{x\to 4}...
4
votes
1answer
104 views

Does the limit exist for $y=\sqrt{x}\sin \frac{1}{x}$?

For the graph of $$y=\sqrt{x}\sin \frac{1}{x},$$ Do the following limits exist? If so, what is it? (a) $\lim_{x \to 0^+} f(x)$ (b) $\lim_{x \to 0^-}f(x)$ (c) $\lim_{x \to 0}f(x)$ By the way, the ...
1
vote
0answers
47 views

Continuity of a function from a pretopological space

It is known that for a function $f$ from a topological space to interval $[0;1]$ to be continuous, it is enough that preimages $f^{-1}]a;1]$ and $f^{-1}[1;a[$ be open in $[0;1]$ for every $a$ in our ...
1
vote
1answer
33 views

What are the values of a and b so that f(x) is continuous for all x?

So I am trying to find the values of a and b such that f(x) is continuous. I know the definition of continuity but I'm having a few problems. $$ f(x) = \left\{ \begin{array}{lll} \...
0
votes
2answers
30 views

Prove f(M) is a closed interval given f continuous on M into R, M compact and connected.

I have what I believe is a proof for this question however being how short it is, I have my reservations. Define the function f continuous on M into R. M compact and connected. Since M is compact ...
2
votes
1answer
77 views

A topological category which is a subcategory of Set

In category theory it is possible to freely chose what are objects and what are morphisms as long as the definitions fulfills the axioms for a category. Now, I'm trying to construct a natural ...
0
votes
2answers
58 views

Does this category explain continuity?

I started study mathematics 1970 in the University of Stockholm and there where no courses in category theory at that time. The students were supposed to do self studies in category theory (and set ...
0
votes
1answer
54 views

Evaluate the limit of the form $\lim_{h\to 0} \frac{f(x_o+h)-f(x_0)}{h}$

Limits of the form $$\lim_{h\to 0} \frac{f(x_o+h)-f(x_0)}{h}$$ occur frequently in calculus. Evaluate this limit for the given $x_0$ and function $f$: $f(x)=3x-4, \ \ \ \ \ \ x_0=2$ Okay so I know ...
0
votes
3answers
46 views

Differentiability of function with respect to its continuity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$, with $$ f(x)=\begin{cases} \dfrac{1-e^{-x}}{x}, & x<0\\[4px] a, & x=0\\[6px] \dfrac{\ln(1+x)}{x}, & x>0 \end{cases} $$ where $a\in\mathbb{R}$...
1
vote
0answers
24 views

Maps of profinite sets

I was trying to prove (or disprove) whether or not all maps between profinite spaces are continuous. One proof in favor was the following. Suppose we have a map of profinite sets $X\to Y$. For any ...