1
vote
3answers
17 views

Negation of continuity applied to a sequence

Show that if it is not true that $\lim_{x \to a} f(x)=l$ then $\exists$ $\epsilon$>0 and a sequence $(x_{n}) \rightarrow a$ as $n \rightarrow \infty$ such that $|f(x_{n})-l| \geq \epsilon$. Now ...
0
votes
2answers
23 views

Continuous Function on closed interval

I am having trouble understanding what this question is asking , by "$f$ has a zero" does it mean "there exists $x$ $\in$ $[a,b]$ such that $f(x)$=$0$? any help on how to answer this question in both ...
-1
votes
0answers
40 views

Study the continuity of a function [on hold]

Determine if the following function is continuous $$f(x) = \lim_{n\to\infty}\frac{x}{(1+2\sin x)^{2n}}$$
0
votes
2answers
29 views

Continuity of 1/x

I am confused with what $8(ii)$ wants from me, I answered the first part of this question with help from the question posted here Is $f(x)=1/x$ continuous on $(0,\infty)$? But the this proves ...
0
votes
1answer
22 views

Proving Uniform Continuity using Bolzano Weierstrass

I have been working on this question for some times, and can't seem to put together the contradiction needed using Bolzano. any help would be greatly appreciated,
0
votes
1answer
22 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
2
votes
1answer
59 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
1
vote
1answer
38 views

If $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$

If a mapping $f:[0,1] \to [0,1] $ is continuous, $f(0) =0 $, $f(1) = 1$, and $f^n(x) \triangleq f \ \circ \cdots \circ f(x) \equiv x $, then $f(x) \equiv x$ The mapping $f$ is injective as $f(x) = ...
0
votes
1answer
37 views

Help, check the uniform continuity

(1) $f(x)=sin(1/x)$ on $(0,1]$ ? ( I know it is not uniform continuous on $(0,1)$) (2) $f(x)= xsin(1/x)$ on $(0,1]$? (3) $f(x)=sin(x^2)$ on $[0, \infty)$?
0
votes
0answers
23 views

Some Continuity Question

Suppose $f(x)$ and $g(x)$ are continuous functions on $[a,b]$ with $f$ monotone increasing. Assume there exists a sequence $x_n \in [a, b]$ such that for all $n \in \mathbb{N}$ , $g(x_n) = ...
0
votes
1answer
22 views

Continuity Function Problem

Suppose f(x) is a continuous function from [0,1] into [0,1]. Show that there exists a point $\xi \in [0,1]$ such that $f(\xi) = \xi$.
0
votes
3answers
45 views

Show a function is not continuous

let $g(x) = x - \lfloor{x}\rfloor$ and I want to show that the function is not continuous. I want to use this definition im pretty sure: "For every open set U in $R$, $f^{-1}$ U is open" But I am ...
1
vote
1answer
24 views

Questions on Continuous Function

I know that it is very obvious that intuitively, a continuous function cannot have any gap in between. However, I am having difficulty proving it. Normally, in textbook and also in my real analysis, ...
0
votes
0answers
37 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
1
vote
1answer
18 views

continuous functions on metric space

Assume $f:X\rightarrow Y$, where $X$ and $Y$ are two metric spaces. If $f(\overline{E})\subset \overline{f(E)}, \, \forall E\subset X$, then how can we prove that $f$ is continuous? Thank you for ...
3
votes
0answers
24 views

Surjective function on a compact metric space [duplicate]

Assume $f:K\rightarrow K$, is surjective and $K$ is a compact metric space and we have $d(f(x),f(y))\leq d(x,y)\, \forall x,y\in K$. How can I prove that $d(f(x),f(y))= d(x,y)\, \forall x,y\in K$? ...
0
votes
2answers
26 views

Continuity of an operator in $C^0[0,1]$ with different norm

Let $C^0[0,1]$ be the space of real valued continuos functions with the norm $\|f\| = \int \limits_{0}^1 x^2 |f(x)| dx$ and let $T \colon C^0[0,1] \to C^0[0,1]$ such that $f(x) \mapsto f(1-x)$. Is $T$ ...
0
votes
1answer
22 views

Discontinuity of the indicator function

Consider the function $q(x,\theta)=1\{ x \in \{x \text{ s.t. } \theta+x_i>0 \text{ }\forall i \}\}$ where 1 is the indicator function taking value 1 if the condition inside $\{ \}$ is satisfied and ...
2
votes
1answer
38 views

A basic confusion over uniform continuity

Suppose $F$ defined on $[a,b]$ is continuous. Is this true that $$ \sup_{0 < h < \frac{1}{n}} \frac{F(x+h) - F(x)}{h} \leq \sup_{h \in \text{rationals between 0 and 1/n}} \frac{F(x+h) - ...
0
votes
2answers
20 views

An MCQ question on continuity.

Let f: R->R be a continuous bounded function, then : A. f has to be uniform continuous. B. There exists an x in R such that f(x)=x C. f cannot be increasing. D. lim x->inf f(x) exists. A ...
0
votes
0answers
10 views

Let C(R) be all real valued continuous functions on R such that limx->+-infinity f(x)=0 show that C(R) is complete with respect to the uniform metric.

I have looked through many other proofs online but none seem to provide a general proof for this they all seem specific to the interval [0,1] and those that are not do not seem to explicitly prove it. ...
0
votes
2answers
65 views

Derivative of $x^2\sin(\frac{1}{x})$

I was reading an article in American Mathematical Monthly and came across this example.It says that derivative of $x^2\sin(\frac{1}{x})$ takes on all values in $[-1,1]$ in any interval ...
1
vote
1answer
19 views

Constant extension of locally Lipschitz function outside an interval is Lipschitz?

Let $f$ be a locally Lipschitz $C^1$ function defined on $\mathbb{R}$. Define $$g_n(x)=\begin{cases}f(x) &: x \in [-n,n]\\ f(n) &: x \in (n, \infty)\\ f(-n) &:x \in (-\infty, ...
0
votes
1answer
14 views

Differentiability and basic definitions

If $f+g$ is differentiable at $a$, must $f$ and $g$ be differentiable at $a$? If " and $f$ is differentiable at $a$, must $g$ be differentiable at $a$? If $f*g$ is differentiable at $a$ and $f$ is ...
1
vote
1answer
26 views

Show that an integral can be made as small as possible.

Consider a function $\mu(s)$ satisfying the following properties: $\mu(s) \in C^0((0,+\infty))$, $\mu(s) > 0$ and $\mu(s)$ is increasing in $s \in (0,+\infty)$, $\displaystyle \int_0^1 ...
-6
votes
1answer
69 views

Countablity of the set of the points where the characteristic function of the Cantor set is not continous

We are creating the Cantor set typically starting from the interval $[0,1]$ and removing $\frac{1}{3}$ of it like it is described here or here. The problem is to resolve if the set of discontinuities ...
3
votes
2answers
44 views

Examples of Functions

Alright so I am trying to find examples of functions that are differentiable at a point, but not continuous there. Also a function continuous at no point; a function continuous only at one point. ...
-1
votes
0answers
12 views

Lipschitz continuity for a function in $\mathbb{R}^n$

How can I show that a function $ f: \mathbb{R}^n \rightarrow \mathbb{R}^n, f(x) = Ax$ where $A \in Mat_{nxn}(\mathbb{R})$ is Lipschitz continuous?
2
votes
0answers
19 views

Show that uniform continuity implies stochastic equicontinuity

Let $\Theta$ be a metric space and assume it is compact. Let $W_t: \Omega \rightarrow \mathbb{R}^k$ be a random variable for $t\leq T$. Let $m(.,\theta): \mathbb{R}^k\rightarrow\mathbb{R}^s$. Let ...
0
votes
1answer
36 views

Proving definition of limits with definition of continuity and visa versa

That is: Let $f: D \rightarrow \mathbb{R}$. Suppose $x_0 \in D$ is a limit point. Prove $f$ is continuous if and only if $\lim_{x\to x_0} f(x) = f(x_0)$. Also, if $x_0$ is not a limit point, prove ...
1
vote
1answer
55 views

Convolution $f*g$ is continuous

Statement: Let $f,g: \mathbb{R}^d \rightarrow \mathbb{R}$ be Lebesgue measurable functions such that $f\in L^1(\mathbb{R}^d)$ and $g\in L^\infty(\mathbb{R}^d)$. The convolution $f*g:\mathbb{R}^d ...
1
vote
2answers
52 views

$\varepsilon$-$\delta$-definition for continuity of $x^n$

Show that $f:\Bbb R\to\Bbb R,x\mapsto x^n$ with $n\in\Bbb N$ is continuous in $x_0=0$ using the $\varepsilon$-$\delta$-definition. We assume that $$\forall ...
2
votes
3answers
34 views

continuity, rational numbers and real numbers

Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $ f(x) = \left\{ \begin{array}{ll} |x| & \mbox{if $x \in \mathbb{Q}$}\\ -|x| & \mbox{if $x \notin \mathbb{Q}$}.\end{array} ...
1
vote
3answers
46 views

Non-decreasing functions and continuity

I have the following situation: $f\colon\mathbb{R}\to\mathbb{R}$ is a non-decreasing $g\colon\mathbb{R}\to\mathbb{R}$ is defined as $\ g(x):=\lim_{t\to x^+}f(t)$ I have proved that also $g$ is ...
0
votes
1answer
46 views

Differentiability conditions for a piecewise function

So this is an analysis class, and we just started the unit on differentiability -- however I missed the class. Can someone start me off with a good real analysis definition for differentiability of ...
2
votes
1answer
47 views

Where is piecewise dirichlet function with $|x|^2$ continuous or differentiable?

If $|x|^2$ is continuous and differentiable on all of $\mathbb{R}^n$ (already shown differentiability by showing all $n$ of its partial derivatives are continuous), then... Question: For the function ...
1
vote
2answers
31 views

bounded monotone continuous function is uniform continuous [duplicate]

If $f: \mathbb R \to \mathbb R$ is bounded, increasing and continuous. Does $f$ have to uniform continuous? I know the answer is yes if $f$ has domain to be any open interval, say $(0,1)$. But I ...
0
votes
0answers
36 views

Totally differentiable function - definition

I know for a function of several variables, if all partial derivatives exist and they are continuous at and around a point $a$ then the function is totally differentiable at that point. I ...
0
votes
3answers
90 views

Continuity proof for exponential

Show that $f(x) = e^x$ is continuous using the epsilon-delta definition. I can't seem to write down anything meaningful...
0
votes
2answers
77 views

Discontinuity of the characteristic function

Let $A \subseteq \mathbb{R}^n$. Let $f(x) = \chi_A $ be the characteristic function, and put $D = \{ x : f(x) \; \; \text{is discontinuous} \} $. Then $\partial A = D $. MY try: Let $y \in D $. ...
1
vote
0answers
110 views

Continuity of a parametric integral (where the integrated function is discontinuous)

For all $t\in\mathbb{R}$ consider $$F(t):=\int_\mathbb{R}e^{-x^2/2}\log|t+e^x|\,dx \;.$$ I managed to show that $F(t)$ is well-defined and finite for every $t$. I would like to show that $F$ is ...
1
vote
1answer
35 views

Continuity of integral of continuous functions

Let $f\in L^1(\mathbb{R})$. Show that the function $g$ defined on $\mathbb{R}$ by $$ g(x) = \int_{\mathbb{R}} \sin(xy)f(y)dy$$ is well defined and continuous on the real line. So I want to prove ...
0
votes
5answers
90 views

Continuous function $f: \mathbb R \to \mathbb R $ such that the set { $ x \in \mathbb R : f(x)<0$ } is singleton

I am in desperate need of an example of a continuous function (if exists) $f: \mathbb R \to \mathbb R $ such that $ f(x) <0 $ for exactly one $x \in \mathbb R $ ; please help .
1
vote
2answers
61 views

modern analysis: integrals and continuity

Let $$f(x) = \sum_1 ^\infty n*e^{-nx}$$ Where is $f$ continuous? Compute $$\int_1^2f(x) dx$$ I am having trouble proving where $f$ is continuous. For the second part, so far I have been able to ...
1
vote
0answers
48 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
0
votes
1answer
74 views

$\exists f:\mathbb{R}\rightarrow \mathbb{R},$ continuous, non-constant, with uncountably many extrema?

I couldnt think of any; by intuition I don't think any can exist, but I can't figure out how to prove it. If it existed then the set of extrema would have to be uncountable but I think this might ...
2
votes
5answers
95 views

How do I prove that, for this given function, $f$ is continuous at $a$ iff $a=-1$?

I'm given the function: $f(x)=\left\{ \begin{array}{lr} -x & : x \in \mathbb{Q}\\ x+2 & : x \notin \mathbb{Q} \end{array} \right.$ How would I (at least go about ...
1
vote
0answers
66 views

Proving a function is continuous using preimages

I want to prove that f is continuous using the preimages of open subsets here. Never worked with pre images before -- can anyone help? (also would love a good definition of a preimage).
1
vote
1answer
54 views

Bounded Jacobian implies uniform continuity

I am trying to solve the following problems but I am not sure what the difference between the 2 problems is. 1) Prove that is $U = B_r(x)$ (open ball centered at $x$ with radius $r>0$) is an open ...
2
votes
3answers
50 views

Prove that a continuous function has given property

The problem: Suppose $f$ is a continuous function on [0,2] and $f(0) = f(2)$. Prove that there exists $x,y \in [0,2]$ such that $|x-y| = 1$ and $f(x) = f(y)$. Intuitively this makes sense after ...