1
vote
2answers
42 views

If a continuous function is positive at a point, it is also positive in some neighborhood of the point [closed]

Suppose that $f:\mathbb{R}^k\to\mathbb{R}^1$ is a continuous function and that $f(x^*)>0$. Show that there is a ball $B=B_\delta(x^*)$ such that $f(x)>0$ for all $x\in B$.
1
vote
1answer
25 views

Solutions depending on something continuously

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find ...
1
vote
3answers
29 views

Where does my proof of uniform continuity fail?

I am trying to prove that $f:R \to R f(x)=\sin x$ is uniformly continuous. I have said: Fix $\epsilon > 0$ and $\delta=\epsilon$ $|\sin x - \sin y| \le |\sin x| - |\sin y| \le 1 - 1 = 0 ...
0
votes
1answer
33 views

Proving a function is not uniformly continuous.

I am using the definition: $(∃ε > 0)(∀n ∈ N)(∃ x_n, y_n ∈ (0,1])[(|x_n − y_n| < δ_n =1/n) ∧ (|f(x_n) − f(y_n)| ≥ ε)]$ to prove that $1/x^2$ is not uniformly continuous. In the solution I am ...
0
votes
1answer
34 views

Prove something that is differentiable

The question states If g(x) is differentiable, then for any positive integer $n$, $(g(x))^n$ is differentiable and $\frac d{dx}$$(g(x))^n=(g(x))^{n-1}g'(x). $ Where does the continuity of g enter ...
1
vote
1answer
51 views

Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$

$f(x)$ is continuous on $[a,b]$. Now we define a new function $M(t)$, for every $t\in[a,b]$ $$M(t) = \sup_{a \leq x \leq t} f(x).$$ Prove formally that $M(t)$ is continuous on $[a,b]$. (sup = ...
0
votes
3answers
66 views

Fixed point and period of continuous function

Prove/ Disprove: Let $f:(0,1)\to(0,1)$ be such that $|f(x)-f(y)|\leq 0.5|x-y|$ for all $x ,y.$ Then f has a fixed point. 2.Let $f:\mathbb R\to\mathbb R$ be continuous and periodic with period ...
1
vote
2answers
107 views

Introduction to Analysis: The Riemann Integral

The following is a problem from Arthur Mattuck's book, "Introduction to Analysis." Page 265. Assume $f(x)$ integrable on $I$. Prove $F(x) = \int_a^x f(t)\,dt$ is continuous on $I$ How would I ...
0
votes
2answers
111 views

Cardinality of $\{ f\in C'[0,1] : f(0)=0, f(1)=1, |f'(t)|\leq 1 \forall t\in[0,1]\}$… NBHM $2007$

Question is to find : What is the cardinality of the following set : $$A=\{ f\in C'[0,1] : f(0)=0, f(1)=1, |f'(t)|\leq 1 \forall t\in[0,1]\}$$ I would like to see for some time that all $f\in ...
4
votes
0answers
52 views

representation of points of continuity of a function $f :\mathbb{R}\rightarrow \mathbb{R}$

Question is : Suppose $f$ is continuous at $x\in \mathbb{R}$ we need : for given $\epsilon >0 $ existence of $\delta > 0$ such that $|x-y|< \delta$ implies $|f(x)-f(y)|< \epsilon$ ...
-2
votes
3answers
66 views

epsilon-delta proof for continuity if $1/f$

How can I prove that $1/f$ is continuous on $[a,b]$ if $f:[a,b] \rightarrow R$ is continuous on $[a,b]$ and $f(x)$ is never $0$ by an epsilon-delta proof? Thank you.
1
vote
1answer
48 views

Introduction to Analysis: Continuity and Sequences

The following is a Theorem the instructor gave up. Let $f(x)$ be defined for $x \approx a$, and suppose that for all {$x_n$} such that $x_n \rightarrow a, x_n \neq a$, we have $\lim_{x \rightarrow ...
1
vote
1answer
210 views

Prove that f(x)=1/(1+x) is not uniformly continuous

How can I prove that $f(x) = \dfrac{1}{1+x}$ is not uniformly continuous on $(−1,\infty)$. Thank you.
2
votes
1answer
61 views

Proofs involving sequential limit

Let $S$ be the domain of the function $f$. Suppose $S=\left\{\frac{1}{n}: n\in\Bbb N\right\}$. Show $\lim_{x\to0}f(x)=L$ iff $\lim_{n\to\infty}f\left(\frac{1}{n}\right)=L$. Idea: I want to say, let ...
1
vote
1answer
74 views

Introduction to Analysis: Multiplicatively Periodic

I was given this problem. Been stuck on it for a while but I have an idea. The problem reads: Call a function "multiplicatively periodic" if there is a positive number $c \neq 1$ such that $f(cx) ...
0
votes
2answers
153 views

Real Analysis: Continuity of a Composition Function

Suppose $f$ and $g$ are functions such that $g$ is continuous at $a$, and $f$ is continuous at $g(a)$. Show the composition $f(g(x))$ is continuous at $a$. My idea: Can I go straight from definition ...
0
votes
1answer
108 views

Introduction to Analysis: Continuity and Limits

My coworker and I were looking at a problem for our Real Analysis class. It reads: Call a function "multiplicatively periodic" if there is a positive number c $\neq$ 1 such that $f(cx) = f(x)$ for ...
0
votes
4answers
112 views

Proof of continuity [closed]

Let $$f:\mathbb{R}\mapsto \mathbb{R}.$$ Prove that if f is differentiable at a real number c, then f is continuous at c.
0
votes
1answer
109 views

Prove that the function $f(x)=\frac{1}{x}$ is continuous at the point x=2.

I am looking to prove that the function $f(x)=\frac{1}{x}$ is continuous at the point x=2. So we nee that given any $\epsilon>0,\ \exists\delta>0$ so that $|f(x)-f(2)|<\epsilon\\$ whenever ...
1
vote
0answers
48 views

Quantitative Economics: Continuity

How do I prove that $f(x)=e^x$ is a continuous function at the point $x=0$? I understand that anything raised to the $0$ power equals $1$, therefore it is continuous. But I don't know how to write a ...
0
votes
1answer
190 views

Proof of equivalent definitions of continuity of a function

Let $X$ and $Y$ be metric spaces, and $f : X\rightarrow Y$ a function I have to prove: (1) $f : X\rightarrow Y$ is continuous (3) $\forall\,F \subset Y closed: f^{-1}(F) \, is\,closed$ from $(3) ...
5
votes
2answers
128 views

If $f$ and $g$ are continuous, prove $f\circ g$ is continuous.

Suppose that $(X,T)$, $(Y,U)$ and $(Z,V)$ are three topological spaces and that $g\colon X\to Y$ and $h\colon Y \to Z$ are continuous. Prove that $h\circ g\colon X \to Z$ is a continuous ...
1
vote
0answers
76 views

What are some general approaches to proving smoothness?

What are some general strategies for proving that a given function $f(x)$ is smooth (continuous in all orders of derivatives)? What properties of a function are needed to carry out a proof? $f(x)$ ...
0
votes
2answers
313 views

$\epsilon$-$\delta$ proof that $f(x) = x^3 /(x^2+y^2)$, $(x,y) \ne (0,0)$, is continuous at $(0,0)$

I need to prove that $f$ continuous at $(x, y)=(0,0)$ using a $\epsilon$-$\delta$ proof $$ f(x, y) = \begin{cases} \frac{x^3}{{x^2 + y^2}},&(x,y)\neq (0,0) \\ 0,&(x,y) = (0,0) \end{cases} ...
0
votes
1answer
68 views

If I wanted to show that an isometry is always continuous, is this right?

So I need to show that an isometry is always continuous, (from $M$ to $N$ in this case) and my first thought was to show that for some $p,q\in M$, $\exists$ $\varepsilon\gt0$ such that $d_M(p,q)\lt ...
1
vote
2answers
562 views

Evaluation of Derivative Using $\epsilon−\delta$ Definition

Consider the function $f \colon\mathbb R \to\mathbb R$ defined by $f(x)= \begin{cases} x^2\sin(1/x); & \text{if }x\ne 0, \\ 0 & \text{if }x=0. \end{cases}$ Use $\varepsilon$-$\delta$ ...
0
votes
1answer
81 views

Continuity and Open sets

So I am having trouble doing the mapping in this problem. I fail to understand what is being mapped. Are both $\mathbf{x}$ and $f(\mathbf{x})$ in the $U$? I see that I must show $n+1$ here, should ...
1
vote
2answers
178 views

Derivative based on continuity

I have a question about whether I am even close to correct. Let $\mathbb{I}$ and $\mathbb{J}$ be open intervals, and the functions $f:\mathbb{I} \to R$ and $h:\mathbb{J}\to R$ have the property that ...
6
votes
2answers
2k views

Continuity proof.

I want to prove that $\exp x$ and $\sin x$ are continuous. This means I want to show that $$\lim\limits_{x\to a}e^x=e^a$$ $$\lim\limits_{x\to a}\sin x=\sin a$$ for any fixed $a \in \Bbb R$. Then I ...