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)

8
votes
0answers
96 views

Exercise about continuous functions

Consider a continuous function $f \, : \, [0,1] \, \longrightarrow \, [0,+\infty)$ such that $f(0)=f(1)=0$ and : $\forall x \in (0,1), \; f(x) > 0$. I would like to prove that there exist ...
39
votes
3answers
2k views

Why did mathematicians introduce the concept of uniform continuity?

I have solved many problems regarding uniform continuity, but still I can't understand the following: Is there any practical application of this concept, or it is just a theoretical concept? Is there ...
0
votes
0answers
42 views

Checking continuity of $g(x)=\begin{cases} \frac{x^{2}}{1+\sin^{2}\left(\frac{1}{x}\right)}, & x\neq0\\ 0, & x=0 \end{cases}$

Let $$g(x)=\begin{cases} \frac{x^{2}}{1+\sin^{2}\left(\frac{1}{x}\right)}, & x\neq0\\ 0, & x=0 \end{cases}$$ Is g(x) continuous at 0? Does g(x) have continuous first ...
1
vote
0answers
21 views

Continuity of operator.

For a fixed given function $f:[0,1]\times \mathbb{R} \rightarrow \mathbb{R}$ which is assumed to be measurable and bounded (i.e. $f\in L^\infty([0,T]\times \mathbb{R})$). Let $U\subset \mathbb{R}$ be ...
2
votes
1answer
43 views

Is this $\lim \ln(f(x))=\ln(\lim(f(x))$ valid?

Is this mathematically legit? $$\lim_{x\to\infty}\ln(f(x))=\ln(\lim_{x\to\infty}(f(x))$$
1
vote
1answer
21 views

Is the following equivalent to $f$ is discontinuous at $x_0$?

If a function $f$ is defined in a neighborhood of $a \in \mathbb{R}^p$, we say $f$ is continuous if for any neighborhood $B(f(a), \epsilon)$ of $f(a)$ in $\mathbb{R}^m$ there exists a neighborhood ...
5
votes
1answer
87 views

Prove there isn't a continuous surjection $f: [0, 1] \to \Bbb R$ (without compactness)

Definitions: Continuous: A map $f: X \to Y$, where $X$ and $Y$ are topological spaces, is continuous if the preimage in $X$ of any open set in $Y$ is open. Subspace topology: If $(X, ...
1
vote
1answer
21 views

Find values of a, b and c for which $f(x)$ if continuous at $x=0$.

If given function $f(x)$ : $$f(x)= \left\{\begin{matrix} (1+ax)^{1/x} \ \quad ;& x <0 \\ b \ \ \qquad \qquad \quad; & x = 0 \\ \cfrac{(x+c)^{1/3} - 1}{x} ; & x >0 ...
0
votes
1answer
27 views

Finding values of a and b such that the given function is continuous at $ x = \frac{\pi}{4} $ and $ x = \frac{\pi}{2}$ .

Find the values of a and b such that the given function is continuous at $ x = \frac{\pi}{4}$ and $x = \frac{\pi}{2}$. $$f(x)= \left\{\begin{matrix} x + a\sqrt{2} \sin x \ ;& 0\le x < ...
0
votes
0answers
20 views

Locally injective function is globally injective [duplicate]

Let $f:\mathbb R\to \mathbb R$ be a continuous: Is the next statement true? If $f$ is locally injective for every real $x$ then $f$ is globally injective in $\mathbb R$ I think this theorem is true: ...
1
vote
1answer
32 views

Differential operator is not continuous between this metric spaces

Let $\mathbb{D}$ be the set of functions $f:[0,1]\to \mathbb{R}$ of class $C^1$ (differentiable with continuous derivative). Let $\mathcal{C}[0,1]$ be the set of continuous functions in $[0,1]$ ($\to ...
0
votes
2answers
32 views

Find an example of continuous but not increasing function whose inverse function doesn't satisfy the Inverse Function Theorem

I have to find an example of a function $f:[a,b]→R$ which is continuous, but not strictly increasing, such that no inverse function $f^{−1}$ satisfy the property of the Inverse Function Theorem.
0
votes
1answer
19 views

weak-star convergence to Dirac Delta function

Let $Y=C_c((-1,1))$. Let $f_j=2j\chi_{(-1/j,1/j)}$, $f_j:(-1,1)\to\mathbb{R}$. Let $\Lambda_j:Y\to\mathbb{R}$ defined by $\Lambda_j(g)=\int f_jg$ and $\Lambda: Y\to\mathbb{R}$ defined by ...
3
votes
3answers
67 views

How do I rigorously show that $f(x, y) = \frac{x}{2|x|\sqrt{|x|+|y|}}$ is continuous when $x, y \neq 0$?

For the function $f : \mathbb{R}^2 \to \mathbb{R}$ to be continuous, I need to show that for some given $\epsilon > 0$, there exists a $\delta > 0$ so that if $||z - z'|| < \delta$, then ...
3
votes
1answer
30 views

Difference between continuous and uniformly continuous functions on a dense metric subspace.

Let $X$ be a dense subset of metric space $(\tilde X,d)$. Let be $(Y,d')$ be a complete metric space and $ f: X \rightarrow Y$ a continuous mapping. It follows from density that for all points in ...
1
vote
3answers
31 views

Continuity definition of a functional

I'm having a hard time understanding the formal definition of continuity of a functional. I'm not sure if such questions are appreciated on this site; so let me know. Definition: The functional ...
3
votes
0answers
44 views

When does the quotient metric is equivalent to the quotient topology?

Suppose that we have an equivalence relation $\sim$ in a topological metrizable space $(X,d).$ Then we can endow $X/\sim$ with the quotient topolgy. Also, under certains circunstances, there exists a ...
1
vote
3answers
39 views

On a question about finite metric spaces

Let $(X,d)$ be a metric space such that every continuous function $f:X\to \mathbb R$ has a finite Image. prove that, $X$ is finite. I tried this: Let $x_0$ be arbitrary element of $X$ and define: ...
1
vote
2answers
44 views

a discontinuous function the square of which is continuous

give an example of a discontinuous function the square of which is continuous. The domain is $[0,1]$. I tried to use the indicator function of rationals, but its square is not continuous. EDIT:I am ...
1
vote
0answers
17 views

Prove that the “additive” operation of the module($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) is continuous.

Consider the following module $\mathcal{M}=$($\mathbb{Z}_{p}^{*},\mathbb{Z}_{p-1},\cdot$) in which the "additive" operation is defined by normal multiplication in $\mathbb{Z}_{p}^{*}$ and scalar ...
-1
votes
1answer
52 views

Check whether the integrand is continuous when evaluating improper integrals

In order to evaluate improper integrals, I need to know whether the integrand is continuous between the limits of the integral. For the lower and upper limits, I believe you find out if it's ...
1
vote
2answers
47 views

Why the continuity of $f$ is not a necessary condition?

I am quite new to functions and continuity, and now I am reading the slides regarding the intermediate value theorem, which is related to continuity of functions. While reading, I found the ...
1
vote
1answer
37 views

What does continuity of a function mapping a topological space to a real line interval mean?

It makes sense for continuity to be defined on a function mapping a real line to a real line. Or how continuity is defined on a function between two topological spaces (every preimage of an open set ...
0
votes
1answer
18 views

Extending the definition of curve length

I know for continuously differentiable curves on closed interval $[a,b]$, the curve length is given by $\Lambda (\gamma)=\int_a^b |\gamma^{'}(t)|dt$. But what about curves such that $\gamma^{'}(t)$ is ...
1
vote
2answers
32 views

Construction of continuous function

Suppose a function $\phi:[0,1] \rightarrow [-1,1]$. Assume that the function $\phi$ has discontinuity at $x=1$ and $\phi(1)=0$. Question: Is it possible to construct a bijection and continuous ...
0
votes
2answers
27 views

On convergent sequences

Suppose that i have and open and surjective map between two metric spaces $\pi\colon X\to Y,$ and a sequence $(x_n)_{n\in \mathbb{N}}$ such that its image by $\pi$ converges. Is it true that ...
0
votes
1answer
46 views

Evaluating $\lim_{n\to\infty}\int_0^1x^nf(x)\,dx$. [duplicate]

Let $f$ be a continuous function on [0,1]. Evaluate $$\lim_{n\to \infty} \int_0^1 x^nf(x)dx$$ My approach : Consider $\int x^nf(x)dx = \frac{f(x)x^{n+1}}{n+1} - \frac{1}{n+1}\int x^{n+1}f(x)dx$ ...
0
votes
1answer
77 views

Spivak's calculus: Chapter 7 problem 18 d)

In cases (a) and (c) [where it was proven that such a number exists for a continous $f$ on $\textbf{R}$], let $g(x)$ be the minimum distance from $(x,0)$ to a point on the graph $f$. Prove that ...
0
votes
1answer
19 views

Monotone functions and distribution functions

I found this quote in a textbook on measure theory I'm studying: Let $f:[a,b] \to \mathbb{R}$ be an increasing function. Since $f$ has only countably many discontinuities, we may assume without ...
1
vote
2answers
50 views

Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
1
vote
1answer
36 views

Characteristic function approximated by continuous function

I am trying to do the following problem Let $E \subset \mathbb R^d$ be measurable and let $\epsilon>0$. Show that if $A \subset E$ is measurable, then there is $f:E \to \mathbb R$ continuous such ...
0
votes
0answers
18 views

“root” of a right-continuous function

Suppose $f:[0,1] \longrightarrow [-1,1]$ is a right-continuous function such that $f(0) < 0$, $f(1) > 0$, and $f$ only changes sign once in the interval $[0,1]$. Suppose we define the "root" of ...
0
votes
1answer
29 views

Continuity and Directional Derivatives

Does every absolutely continuous function on a compact set possess a left and right hand derivative everywhere on its interior? Although the two need not be equal of course.
2
votes
1answer
42 views

Does the function $f(x) = \frac{x^2-1}{x-1}$ have any point discontinuity?

Since the domain of $f(x)$ is $(-\infty, 1) \cup (1, \infty)$ is there any point discontinuity in $f(x)= \frac{x^2-1}{x-1}$?
23
votes
5answers
1k views

What is the geometrical difference between continuity and uniform continuity?

Can we explain between ordinary continuity and Uniform Continuity difference via geometrically? What is the best way to describe the difference between these two concepts to someone else? Where the ...
3
votes
1answer
45 views

Show that $f$ is continuous at exactly one point

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)= \begin{cases} 5x+7 & \text{ if } x \text{ is rational } \\ x+11 & \text{ if } x \text{ is irrational } \end{cases}$$ ...
0
votes
1answer
61 views

If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
2
votes
2answers
45 views

Continuous function without a weak derivative

Let $f:\Omega\to\mathbb{R}$ be a continuous function. Is it necessarily true that $f$ has a derivative in the weak sense? That is, is there some $v:\Omega\to\mathbb{R}$ such that for every test ...
0
votes
0answers
14 views

continuity and convergence [duplicate]

If we have a continuous function that converges on a compact subset of a metric space does it imply that it converges uniformly in general, or is this only in the case if f is monotonic (Dini's ...
1
vote
2answers
69 views

Convergence of a sequence of integrals

I've tried expanding the hinted expression by using the definition from part (i) and choosing an X0 sufficiently large that |f(x)-l| < 1 but this doesn't appear to help very much at all. I've ...
1
vote
2answers
36 views

Continuity and measurability

this question concerns continuity and measureability. Am I right in thinking that if $f>0$ for all x then $\log(f(\lambda x)/f(x))$ is a continuous function for all $x$. Does this then mean that ...
1
vote
2answers
29 views

Continuity of $\mu(t)=\inf\{x \in \mathcal C : \kappa(x)=t\}$.

Let $\Delta = \{ 0, 1\}^{\mathbb N}$ be a Cantor set. Define $\theta : \Delta \to [0,1]$ by the formula $$\theta(x_1,x_2,\dots) = \sum_{n=1}^\infty \frac{2x_n}{3^n}.$$ Denote $\mathcal C = ...
3
votes
0answers
44 views

Continuity of a function in a locally convex topological space

I endow the space of bounded sequences with a locally convex topology $\tau$ such that $\tau$ is strictly finer than the product topology (the topology of pointwise convergence), $\tau_p$, and ...
0
votes
1answer
20 views

Showing that the functional $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$ is continuous

Suppose that we have the functional $L: L^2[a,b] \to \mathbb{R}$ , $L[h(x)]=\int_{a}^{b}h(x)f(x)dx$. $f(x)$ is a well behaving, integrable function in $L^2[a,b]$. I want to show that this is a linear ...
0
votes
1answer
21 views

Can unbounded discontinuous functions be locally bounded?

Consider the function $$f(x) = \frac{x^3}{1+x^3}$$ Obviously this function is discontinuous at $x = -1$ therefore discontinuous on $\mathbb{R}$. Moreover, it is unbounded at the same point. Now, I ...
1
vote
1answer
38 views

Two functions equal in some point

I have two continuous functions $f,g$, $f(0) \lt g(0), f(1) \gt g(1)$. How do I prove without using "advanced" theorem (using only definitions of limit, continious functions and sup/inf definitions), ...
3
votes
2answers
31 views

Rationale behind a proof regarding a continuous function and an open ball

can I have the rationale for the first line of this proof? i.e. How did you know to start answering the question in this manner? I am guessing it is because you want to exploit the definition of ...
0
votes
1answer
21 views

On the definition of uniform continuity over an interval.

I was reading some slides and I stumbled upon this definition of uniform continuity in an interval I am unsure on how to trace this back to the definition of uniform continuity that I know: A ...
1
vote
1answer
25 views

Proving isometry and continuity from a positive definite symmetric real matrix

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 $\epsilon$ be the Euclidean metric on $\Bbb R^n$ ...
-1
votes
0answers
23 views

Continous frunctions problem

The problem says: f,g:[0;1]->[0,1] ,2 continous functions.They have the property that f(g(x))=g(f(x))). To solve: Both having the property of DARBOUX on the interval ,demonstrate that the numbers "c" ...