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)

1
vote
0answers
28 views

Proof strategy: continuity of an integral

Consider $g: I \rightarrow \mathbb{R}$ given by $$g(x) = \int_{x_0}^x f(y)dy$$ If $f : I \rightarrow \mathbb{R}$ is Riemann-integrable, then $g$ is continuous on $I$. Proof: For any $x_1 \in I$, ...
5
votes
1answer
107 views

When does continuous (or uniformly continuous ) function between normed linear spaces carries bounded sets to bounded sets ?

I know that if $f:\mathbb R^m \to \mathbb R^n$ is continuous then $f$ carries bounded sets to bounded sets . What if we say $X,Y$ are normed linear spaces and $f:X \to Y$ where $f$ is continuous ? ...
10
votes
1answer
158 views

If $f:X \to X$ is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous?

If $f:X \to X$ (codomain and domain have the same topology) is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous? Note that the orbit being finite and $f$ being a ...
1
vote
1answer
40 views

Proof of continuity of all functions on N

My task is to basically proof that any function defined on $\mathbb N$ is a continuous function. I wanted to use the definition that states that if $f$ is continuos at every point a in the domain ...
0
votes
2answers
25 views

Example is required

I am trying to find a seuqence of a continuous functions $\{f_n\}$ defined on $[0,1]$ bounded by some small number, say $\varepsilon$ with the additional requirement of $f_n^\prime(t_0)=1$ at a ...
0
votes
0answers
18 views

The definition of $C(\bar U)$.

In 'Partial differential equations, Evans', $C(\bar U)$ is defined by the space of continuous functions $u\in C(U)$ such that $U$ is uniformly continuous on bounded subsets of $U$. But I have known ...
3
votes
0answers
25 views

Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] ...
0
votes
0answers
15 views

How do I prove or disprove $h$ is continuous at $0$? [duplicate]

I did a problem similar to this where we use a sequence ($x_n$) = $\frac{1}{\sqrt{2n\pi + \frac{\pi}{2}}}$. And then we take the limit? Can someone help me out here?
2
votes
1answer
27 views

What is K-Lipschitz and how do I use it to prove this problem?

So, I'm completely lost here. Can someone explain to me what K-Lipshitz is? And how I'm supposed to prove this problem?
1
vote
0answers
19 views

Connection between Darboux property and semicontinuity

Is there a connection between the Darboux property (that is, the mean value propriety) and semicontinuity? That is, is there a characterization of semicontinuity that uses Darboux property (or are ...
0
votes
1answer
23 views

Let $f:\mathbb{R}→\mathbb{R}$ be defined by $f(x)=\sqrt[3]{x}$. Use the definition of continuity to prove that $f$ is continuous at 0?

Let $f:\mathbb{R}→\mathbb{R}$ be defined by $f(x)=\sqrt[3]{x}$. Use the definition of continuity to prove that $f$ is continuous at $0$? How am I supposed to do this? Can someone please help ...
1
vote
0answers
33 views

If a function is continuous at $x_0$, then it must be defined on a neighborhood of $x_0$?

The title is all my question. Let me state it again: If a function is continuous and defined at $x_0$, then it must be defined on a neighborhood of $x_0$? It seems trivial, but I cannot prove ...
2
votes
1answer
55 views

If a continuous function is never $0$, it must either be always positive or always negative

Let $I$ be an interval, and let $f : I \to \mathbb{R}$ be a continuous function on $I$. Suppose $f$ is never $0$ on $I$. Prove that $f$ must either be always positive or always negative. I was ...
0
votes
5answers
43 views

Is continuity the same as domain being all real numbers?

Basically, my question is— Is this statement: f(x) is continuous for all x. the same as? The domain of f(x) is all real numbers.
3
votes
1answer
25 views

Continuity and maxima and minima [closed]

Is $\sin (t)/t$ continuous at $t=0$? And also, if a function $f(x)$ is of indeterminate form at $x=a$, can it be continuous if $f(a)$ does not exist? Can a discontinuous function have a local maximum ...
0
votes
1answer
55 views

Continuity of a function on the complement of a set of Jordan measure zero

Let $f:D\subset \mathbb R^n \to \mathbb R$ $f= \begin{cases} c, & \text{if $\vec x \in \Omega$} \\[2ex] 0, & \text{if $\vec x \notin \Omega$} \end{cases}$ where D is a closed rectangle and ...
1
vote
2answers
33 views

Computing the limit of an integral (Derivatives of Integrals)

Assuming that $f(x)$ is continuous in the neighborhood of $a$, compute $$ \lim_{x \to a} \frac{x}{x-a} \int_a^x f(t)dt $$
0
votes
1answer
36 views

Difference between definition of a limit if a function and definition of continuity.

I am having trouble understanding why this particular difference exists between the definition of a limit of a function and definition of continuity. Heres the definition of a limit of a sequence. ...
0
votes
0answers
36 views

Multivariable: Continuity of Piecewise function

I have this Multivariable problem.... Where I have to find out if a function is continuous or not. Here is the problem: $f(x, y)=\left\{\begin{matrix} \frac{x^4+3y^4}{x^2+y^2} & (x,y)\neq (0, ...
1
vote
1answer
35 views

Use the definition of differentiation on a piecewise function.

I need to find the derivative at $x=0$. $$ f(x)= \begin{cases} x^2\sin(1/x) & \text{if } x\neq 0 \\ 0 & \text{if } x \leqslant 0 \end{cases} $$ Using the definition, I know that it's equal ...
1
vote
1answer
42 views

If $f:\mathbb C \to \mathbb C$ is continuous at a point $z_0$, then show that $\overline {f(\bar z)}$ is also continuous at $z_0$.

If $f:\mathbb C \to \mathbb C$ is continuous at a point $z_0$, then show that $\overline {f(\bar z)}$ is also continuous at $z_0$. Is the same true for the differentiability at $z_0$? I'm trying to ...
1
vote
2answers
26 views

Continuity and integrability;is it true?

If we have a discontinuous real function of all nonnegative terms and $\int^b_a fdx=0$, then does that necessarily imply $f(x)=0$? I can't come up with an example to help me understand.
0
votes
1answer
40 views

Discontinuity and differentiation;is this possible?

If $ f $ is a continuous function defined on a real interval that has a discontinuity at a point (but is continuous otherwise), then is it possible to be differentiable at that point?
0
votes
0answers
22 views

Rotations of the Plane

When is a rotation of a plane not continuous? I know that if I take a point $(x,y)$ then the rotation is $(x\cos(\theta)-y\sin(\theta),x\cos(\theta)+y\sin(\theta))$, but I keep getting that this ...
3
votes
0answers
33 views

Can we find such a monotone function? [duplicate]

Can we find a monotone function $f:[0,1]\rightarrow\mathbb R$ whose discontinuity set is exactly the set $\mathbb Q\cap [0,1]$? Or can we prove that such a function does not exist?
1
vote
0answers
51 views

Does having infinite limits at a point of discontinuity imply having a vertical asymptote?

Considering that discontinuities occur at holes, jumps, and vertical asymptotes. Is it possible for a function to have a limit from the left of infinite and the limit from the right - infinite if the ...
1
vote
1answer
40 views

Compute the righthand limit; calculus

Let $ f $ be a function defined on a real interval from $0 $ to $1$ and have a discontinuity at $1/2$ (however the righthand and lefthand limits still exist). Let $ F $ be a function defined by $ ...
1
vote
0answers
22 views

Continuity of Product Topology [duplicate]

Let $X_1, X_2, Y$ be topological spaces and let $X_1 \times X_2$ be the topological space obtained by furnishing the Cartesian product set with the product topology. Let $f: X_1 \times X_2 \to Y$ be a ...
0
votes
3answers
90 views

Prove that a function that maps a discrete metric space to any metric space is continuous [closed]

Let $f:D→M$ where $M$ can be any metric space and $D$ is any set with the discrete metric. Prove that $f$ is continuous. I'm not sure where to begin with this.
4
votes
1answer
432 views

Given two potatoes, prove that there is a loop of wire which fits around both

This is a classic problem in geometric continuity and I want to see if there are some solutions other than the one I'm thinking of: Two potatoes are given. Prove that there exists a closed loop of ...
1
vote
2answers
254 views

Definition of sequential continuity: converse?

A function $f: \mathbb R \to \mathbb R$ is called sequentially continuuous if $x_n \to x$ implies $f(x_n) \to f(x)$. Every continuous function is sequentially continuous. Let $f$ be a continuous ...
0
votes
1answer
27 views

The definition of continuity for linear functionals

I am trying to prove that a linear functional is continuous on the space $H^1(0,l)$, and I have a couple of different definitions. The one that I want to use is that $f$ is continuous if $f$ is ...
1
vote
2answers
27 views

Continuity of an increasing function on a dense set

Let $f$ be increasing on $D$ ($D$ is dense in $\mathbb{R}$), and define $\tilde{f}$ on $(-\infty,\infty)$ as follows: $$ \forall x: \tilde{f}(x) = \inf_{x<t\in D} f(t).$$ Show that continuity of ...
1
vote
1answer
42 views

Problem in standard proof of continuity when pre-image is open?

I have seen several proofs of the fact that a function $f$ from a metric space $X$ to a metric space $Y$ is continuous if every open set on $Y$ has an open inverse image on $X$. When proving the ...
1
vote
0answers
32 views

If f is continuous and f $\in$ L1 then $\lim_{\tau\to \tau_0} \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $0$?

Where $ \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $\int_\mathbb{T} \vert f(t-\tau)-f(t-\tau_0)\vert \ dt$ I find it easy to see when f is uniformly continuous, since we would have $\vert ...
0
votes
1answer
92 views

Differentiable implies continuous - in more dimensions?

I know that "differentiable of function f in $x_0$ implies continuous of function f in $x_0$". Can I use the same proof to show that it is valid for a function $f: M \to \mathbb{R}^m$ with $M ...
0
votes
0answers
20 views

Differentiability of a upper semicontinuous function

assume I am facing the following function: $$f(x)=ln(x)+\imath_{x\ge y}$$ It is clear that it is upper semicontinuous. But can anyone give me a hint how to see if the derivative is continuous or not ...
0
votes
2answers
125 views

Determining whether a function is uniformly continuous

Determine whether $(4x-3)/(x-2)$ is uniformly continuous on the open interval $(1,2)$. I'm not sure how to start this as I have only answered these questions with closed intervals?
1
vote
1answer
45 views

Closure of the interior of the epigraph

Suppose $f:E\to(-\infty,\infty]$, where $E$ is a Banach space, is lower semi-continuous, convex, and the interior of $epi(f)\neq\emptyset$. Show that $\overline{int(epi(f))} = epi(f)$ \begin{equation} ...
0
votes
1answer
42 views

Prove $f(x) = 1/x$ is continuous at $x = 1/2$

I need to write an $\epsilon, \delta$ proof. I know that $\delta$ must be less than $\frac12$, but I can't figure out the other $\delta$ in terms of $\epsilon$. Thank you in advance!
0
votes
2answers
44 views

Proving continuity of $x^{2}-2$ with $\epsilon$ and $\delta$

Here is my attempt: $f$ is continuous in $p$ if $\forall \epsilon > 0\; \exists \delta > 0$ such that $|x-p|<\delta \Rightarrow |f(x) - f(p)|<\epsilon$ Then for $f(x)=x^{2}-2$ in $x=p$ ...
0
votes
1answer
25 views

Proving extreme value theorem; is showing maximum enough?

Can we prove the extreme value theorem by merely showing that a maximum exists (if $f$ is continuous and defined on a closed, bounded interval in $\mathbb{R}$) because then we'd apply this "half" of ...
0
votes
3answers
39 views

Prove or disprove: If $f$ is continuous and differentiable in $[a,b]$ then $a$ is a local minimum or a maximum point in $[a,b]$.

Prove or disprove: If $f$ is continuous and differentiable in the interval $[a,b]$ then $a$ is a local minimum or a maximum point in $[a,b]$ I'm trying to disprove by giving a counterexample, any ...
-3
votes
1answer
31 views

Examine the continuity of complex function

There is confusion regarding continuity of the following function. When solving in polar form it comes continuous but when solving in $x$ and $y$ then not continuous. Examine the continuity of ...
3
votes
2answers
70 views

Existence of Zero Divisors in $C(X,\mathbb{R})$

Consider any topological space $X$ and $\mathbb{R}$ be with usual topology. The set of all continuous functions from $X$ to $\mathbb{R}$, denoted by $C(X,\mathbb{R})$, is a commutative ring with unity ...
0
votes
0answers
24 views

How to prove that a function's image is closed and bounded, without using Heine-Borel's theorem?

Just wanna apologize for potential mistakes since I've never asked any questions here. I've been trying to solve this one all day long, but did not succeed. I have a continuous function and it's ...
0
votes
2answers
67 views

Prove: using the Fundamental theorem of calculus

I have trouble doing proofs using the fundamental theorem of calculus and I think seeing an example would help. Suppose we have a continuous function $ f $ defined on a real interval and a function $ ...
1
vote
0answers
28 views

Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
2
votes
1answer
26 views

How to determine continuity in higher dim

$$f(x,y) = \frac{1-\cos{\sqrt{xy}}}{y}$$ $$f(x,0) = \frac{x}{2}$$ How do I prove this is continuous in the quadrant $x,y \ge 0 $? I can't find counterexamples (weak). I'm just starting working in ...
1
vote
1answer
46 views

If $f$ is uniformly continuous on $[1,\infty)$ then $\exists \lim_\limits{n\to \infty}{f(x)}=L$, $-\infty \le L\le \infty$.

Prove or disprove: if $f$ is uniformly continuous on $[1,\infty)$ then there exists a limit, $\lim_\limits{n\to \infty}{f(x)}=L$ where $-\infty \le L\le \infty$. I tried to find a counterexample but ...