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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3
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2answers
26 views

Show that $f(x):=\frac{2x^3+x^2+x\sin(x)}{(\exp(x)-1)^2}$ is continuously extendable to $x_0=0$.

What I know If $\lim\limits_{x \to x_0}f(x) := r$ exists, we can create a new function $\tilde f(x) = \begin{cases} f(x) &\text{if }x\in\mathbb{D}\setminus x_0 \\ r & \text{if }x = x_0 ...
-1
votes
1answer
57 views

What type of discontinuity is found in this graph?

$$ f(x) = \begin{cases} \dfrac{1}{x} && \text{when $x > 0$}\\ 4 && \text{when $x < 0$} \end{cases} $$ What type of discontinuity is present when $f(0)$ ? ...
1
vote
2answers
16 views

If $g(t)=d$ (constant), show that all solution of $L[y] \to \frac{d}{c}$ when $t \to \infty$

Let the differential equation $L[y] = a y'' + by' + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $g(t)=d$ (constant), show that all solution of $L[y] \to \frac{d}{c}$ when $t ...
0
votes
1answer
15 views

Show that $Y_1[t]- Y_2[t] \to 0$ as long as $t \to \infty$ - Differential equations

Let the differential equation $L[y] = a y'' + by' + cy = g(t)$, where $a$, $b$ and $c$ are strictly positive numbers. If $Y_1(t)$ and $Y_2(t)$ are solutions at the $L[y]$ equation, show that $Y_1[t]- ...
0
votes
0answers
8 views

Square wave in the limit of infinite frequency

What is the new function obtained when one takes the limit of the square wave function when the frequency is taken to infinity? Does it depend on how the function is written down (e.g. defined as ...
0
votes
1answer
24 views

Strictly monotone functions and continuity

Let $f : X \to I$ be a strictly monotone surjective function mapping $X \subseteq \mathbb{R}$ to an interval $I \subseteq \mathbb{R}$. Then is $f$ necessarily continuous? Without loss of ...
-1
votes
1answer
21 views

Investigation of continuity of a piecewise function [on hold]

Investigate continuity of the following piecewisely defined function: $$f(x)= \begin{cases} x & \text{if } x \in \mathbb{Z} \\ 0 & \text{otherwise} \end{cases}$$ where $\mathbb{Z}$ is the ...
1
vote
1answer
6 views

Does continuity in one variable and locally Lipschitz in another imply uniformity in the first?

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $ f(t,x):D ...
-1
votes
1answer
28 views

Find the values of constants in piecewise [on hold]

Find the value of the constants a and b so that the function defined by $$ f(x) = \begin{cases} x+1 ,& 1<x<3 \\ x^{2}+bx+c, &|x-2| \geq 1 \end{cases} $$ is continuous in ...
2
votes
1answer
38 views

composition of functions is continuous

Question is as follows : Let $X,Y,Z$ are metric Spaces Let $f:X\rightarrow Y$ be continuous map onto $Y$ and let $X$ be compact. Also $g:Y\rightarrow Z$ such that $g\circ f:X\rightarrow Z$ is ...
0
votes
1answer
38 views

Let $f:(0,1)\to (0,1)$ be a continuously differentiable function. Then which of the following are true?

Let $f:(0,1)\to (0,1)$ be a continuously differentiable function. Then which of the following are true? $1)$ $g=1/f$ is continuous function on $(0,1)$. $2)$ $g=1/f$ is continuously ...
1
vote
1answer
86 views

Let $f$ be injective and discontinuous at some point $c$. Can its inverse be continuous?

$f$ is injective at an interval $[a,b]$, but discontinuous at some point $c$ in the same interval. I need to prove that its inverse is continuous at that interval. Should I consider what is the ...
0
votes
2answers
45 views

Let f be continuous. By EVT there exists a c such that f(c)=supx f(x). Show that f is not injective.

I am given a continuous function f in an interval [a,b]. To show that f is not injective, should I consider the definition of the extreme value theorem? I am not sure how to show that it is not one ...
0
votes
0answers
4 views

Continuity of utility function in normal form games

I want to characterize the utility functions of normal form games. Let $G$ be a game with a finite number of players $k$ given by the action sets $S_1,\ldots,S_k$ and utility function $u:S_1\times ...
1
vote
0answers
13 views

$A,B$ closed subsets of $\mathbb R^n$ , when can we say (other than compact-ness of $A$ or $B$ ) $\exists b \in B$ such that $dist(A,B)=dist(b,A)$ ?

Let $A,B$ be disjoint closed subsets of $\mathbb R^n$ , when can we say ( weaker than compact-ness of $A$ or $B$ ) that there exist $b \in B$ such that $dist(A,B)=dist(b,A)$ ? I know that if $A,B$ are ...
-1
votes
1answer
42 views

CALCULUS: Sketching a function by given conditions [on hold]

Pls help. I'm currently on a struggle with this calculus problem. Thanks in advance.
1
vote
0answers
27 views

Discontinuous parametric integral function

Is there an example of a function $f:[0,1] \times [0,1] \to \mathbb{R}$ such that for all $x \in [0,1]$ the function $\phi(y) = f(x,y)$ is continuous in $y$ and for all $y \in [0,1]$ the function ...
0
votes
2answers
32 views

Show that $\langle\cdot,\cdot\rangle : E \times E \to \mathbb{R}$ is a continuous function

Let $E$ a normed vector space, where the norm is induced by a dot product. The norm of $E \times E$ is defined as $||(x,y)|| = \max\{||x||,||y||\}$. Show that $\langle\cdot,\cdot\rangle : E ...
1
vote
1answer
24 views

Proving a norm is lipschitz

Let $M\in\mathbb{R}^{n\times n}$. Define the function $f\colon\mathbb{R}^n\to\mathbb{R}$ by $f(x)=\Vert Mx\Vert$. Show that $f$ is Lipschitz. Let $x,y\in\mathbb{R}^n$, then we want to find a ...
1
vote
0answers
25 views

quotient of two differentiable functions is differentiable

I have two functions $k(t)$ and $l(t)$ in a certain closed interval $[a,b]$ both functions are continuous and differentiable in the interval. In addition we have: Both functions are increasing with ...
0
votes
1answer
34 views

conditions for continuous function

A function $f\colon [0,1]\to [0,\infty)$ is continuous and satisfies $f(0) = \lim_{x\to 0^+}\frac{f(x)}{x}$ und $ f(x)\le\int_0^x \frac{f(s)}{s}ds$ for all $x\in[0,1]$. I'm curious if it implies ...
0
votes
1answer
21 views

Continuity of the function $f(x)=\lim\limits_{n \to \infty}\frac{x}{1+(2\sin(x))^{2n}}$

I was studying the continuity of the function: $f(x)=\lim\limits_{n \to \infty}\frac{x}{1+(2\sin(x))^{2n}}$ I understood that the function behave as $ f(x)=x \quad2 \sin(x) \leq 1 \\ f(x)=0 ...
0
votes
1answer
28 views

Is this operator a distribution?

Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$ a distribution (generalized function)? I need to check two things: whether ...
0
votes
1answer
18 views

How to show map is non-singular

Let $f:\;\mathbb{R}^n\to\mathbb{R}^n$ be differentiable. Suppose that for all $x\in\mathbb{R}^n:$ $$\lVert \mathrm{D}f(x)-\mathrm{I}\rVert\leq \frac{1}{2}$$ where $\lVert\cdot\rVert$ is the ...
1
vote
1answer
40 views

Proving a Function Continuous with Non-Standard Analysis

I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. ...
0
votes
0answers
13 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
-1
votes
1answer
52 views

Finding the domain of $\frac{1}{x}|x^2 - 1|$ [on hold]

What is the domain of this function $F(x)=\frac{1}{x}|x^2 - 1|$ Can someone please tell me how to find it ?
3
votes
3answers
53 views

How to show that $f$ is a straight line?

Let $f:\mathbb R\to\mathbb R$ be continuous such that $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}~\forall~x,y\in\mathbb R.$ How to show that $f$ is a straight line?
1
vote
2answers
66 views

Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
3
votes
1answer
27 views

Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$?

Is it possible to extend $f(z)=\frac{\Re(z)}{|z|}$ by continuity at $z=0$? Let $z=r(\cos(\theta)+i \sin(\theta))$. Then $\frac{\Re(z)}{|z|} = \frac{r \cos(\theta)}{r} = \cos(\theta) $; as the ...
-5
votes
0answers
20 views

Calculus: Proving Continuous Function by Intermediate Value Theorem [duplicate]

Prove step by step: Let $f(x)$ be a continuous function from the closed interval $[a, b]$. Use the Intermediate Value Theorem to show that $f(x)$ has a fixed point, that is, there is a point $x \in ...
1
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0answers
17 views

Existence of a limit - Composition of continuous functions - Questioning [duplicate]

The question of Jim Darson to this link, Don Antonio replied using a similar property in the composite of continuous functions ($\frac{\text{Re}\,z}z$ and the line $\;y=mx\;$) is continuous, but with ...
0
votes
1answer
25 views

Lipschitizianity of the square root of a positive $C^2$ function

I was trying to solve this exercise. Let $f\in C^2(\mathbb{R})$ a strictly positive function such that $f''$ is bounded. Then prove that $\sqrt{f}$ is Lipschitz. A first idea was to prove that it's ...
0
votes
2answers
51 views

Continuity and differentiability for $\sin(\sqrt x)$ & $\sinh(\sqrt {-x})$?

Let $f: \Bbb R \to \Bbb R$ with $$f(x)= \begin{cases} {\sin(\sqrt{x})\over\sqrt{x}},& \text{for } x>0\\ 1,& \text{for } x=0\\ ...
0
votes
2answers
29 views

Is it possible to extend $f$ by continuity at $z = 0$? Why or why not?

Let $f(z) = \frac{z}{|x|}$, with $z \not=0$ (a) Construct two sequences ${u_n}$ and ${v_n}$ such that $\lim_{n \to \infty} u_n = 0$ and $\lim_{n \to \infty} v_n = 0$ $\lim_{n \to \infty} f(u_n)$ ...
4
votes
2answers
515 views

If a function is discontinuous at one point, then filled in, is it now continuous?

I am looking at the continuity of the following function $f(x) = \sin(1/|x|), f(0) = 0$ So this is $f(x) = \sin(1/|x|)$ filled in at $x = 0$ Clearly, $\lim\limits_{x \to 0} f(x) = 0 $ by squeeze ...
1
vote
2answers
33 views

Can I prove a function is continuous by looking at the domain?

I came across the following question in a calculus book: For the function $$f(x)=1-\sqrt{1-x^2}$$ show that it is continuous on the interval $$-1≤x≤1$$ The solution in the book showed that the one ...
0
votes
1answer
21 views

How can I show the points of continuity of the following function

How can I show the points of continuity of the following function $$f(x) = \begin{cases} 2x, & \text{if $x \in \Bbb Q$} \\[2ex] x+3, & \text{if $x \in \Bbb I$ } \end{cases}$$ I am having ...
-3
votes
2answers
60 views

Help with continuity [closed]

Could you please clarify these questions to me. Find all the numbers for which the given function is discontinuous. $F(x)=[x-1]$ I think the solution is $\Bbb Z$ numbers right ? $F(x)= ...
1
vote
1answer
16 views

Nonlinear operator sends bounded set to relatively compact set

Consider $g$ a continuous function on $[a,b]\times\mathbb{R}$, and let $z_0\in\mathbb{R}$. Define the (nonlinear) operator on $C[a,b]$: $$Mv(x)=z_0+\int_a^x g(t,v(t))\,dt$$ for $x\in[a,b]$. Prove ...
0
votes
1answer
31 views

Let $S=[0,1) \cup [2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected. Which of the following statements is true?

$f$ has exactly one discontinuity. $f$ has exactly two discontinuities. $f$ has infinitely many discontinuities. $f$ is continuous. I know theorems related to connectedness and ...
0
votes
1answer
49 views

Why is continuity needed to substitute value of derivative inside Riemann-Stieltjes Integral?

Given $f$ increasing on $[a,b]$, $g(x)\in R(\alpha)$ on $[a,b]$, $\alpha \in C([a,b])$ and $\alpha \in BV([a,b])$ $$ \beta(x)=\int_a^xg(z)d\alpha(z) \text{ on [a,b]} $$ Why is the additional ...
0
votes
2answers
42 views

Show that $f^{-1}$ is continuous

Let $E$ and $F$ two normed vector spaces, $A \subset E$ compact, $B \subset F$ and $f: A \to B$ is a bijective continuous function. As $f$ is bijective, we can defining the inverse function ...
1
vote
2answers
17 views

Show that $Gr(f)$ is compact

Let $A \subset \mathbb{R}^n$ a compact and $f : A \to \mathbb{R}^m$ a continuous function. Let the graph of $f$ $$Gr(f) = \{(x,f(x) : x \in A)\}.$$ Show that $Gr(f)$ is compact. My proof : ...
0
votes
1answer
17 views

Definition of continuity up to the boundary

Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does it mean $f\in C(\bar{\Omega})$, i.e. what does it mean $f$ to be continuous at $x \in \partial \Omega$, maybe $$\forall \epsilon >0 ...
0
votes
2answers
74 views

Is there a nice open set proof that multiplication is continuous?

For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the ...
-1
votes
1answer
40 views

Piece wise function continuity [closed]

Find all values of $a$ and $b$ so that the following function is continuous for all value of $x$. ($x\in\Bbb R$). $$ f(x)=\begin{cases}-3a+4x^5b&\text{when }x\le -1\\ ax-2b&\text{when ...
3
votes
2answers
40 views

Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
0
votes
1answer
18 views

Continuity proving of function with delta-epsilon

Prove continuity of function with the delta-epsilon definition in point $x_o=0$ $$f:\mathbb{R}\rightarrow \mathbb{R}$$ $$f(x) = \begin{cases} x^2+1, & x \in \mathbb{Q} \\[2ex] 2^x, & x \in ...
-3
votes
1answer
49 views

Complex Continuity [closed]

Is the function $f$, defined by $$ f(z) = \begin{cases} \frac{z^2+2iz-1}{2z^2+iz+1} & \text{ if } z \not \in \{-i\}\\ 0 & \text{ if } z = -i \end{cases}$$ continuous at $−i$? Explain your ...