0
votes
2answers
29 views

If f is continuos on an interval, is it then uniformly continuous

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I now know that it is not. Can someone give me a proof ...
0
votes
1answer
24 views

If $f$ is continuous on $(0,5)$, is it uniformly continuous on same interval

I have a function, $f$ that is differentiable on $(0,5)$, and I know it is continuous on $(0,5).$ Is it also uniformly continuous on $(0,5)?$ I believe it is. I now know that it is not. Can someone ...
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 ...
0
votes
0answers
45 views

$f \in \mathcal{R}(\alpha)$ on $[a,b]$, then $\exists P_n$ s.t. $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$.

Assume $f \in \mathcal{R}(\alpha)$ on $[a,b]$, and prove that there are polynomial $P_n$ such that $\lim\limits_{n \rightarrow \infty} \int\limits_{a}^{b} |f-P_n|^2 d \alpha =0$. This is what I have, ...
3
votes
1answer
113 views

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,…)$, prove that $f(x)=0$ on $[0,1]$.

If $f$ is continuous on $[0,1]$ and if $\int\limits_{0}^{1} f(x) x^n dx = 0, (n=0,1,2,...)$, prove that $f(x)=0$ on $[0,1]$. This is what I have, how does it look? Proof: Let $P(x)$ be any ...
0
votes
0answers
87 views

$\{f_n\}$ equicontinuous sequence of functions on compact $K$, converges pointwise on $K$ then converges uniformly on $K$.

Suppose $\{f_n\}$ is an equicontinuous sequence of functions on a compact set $K$ and $\{f_n\}_{n=1}^{\infty}$ converges pointwise on $K$. Prove that $\{f_n\}_{n=1}^{\infty}$ converges uniformly on ...
1
vote
1answer
32 views

Lower semi-continuity of a convex functional on $L^1(\Omega,[0,1])$

Let $\Omega$ be a bounded domain and $f:\Omega\times[0,1]\to[0,\infty]$ be such that $x\mapsto f(x,u)$ is measurable for every $u$, $u\mapsto f(x,u)$ is continuous and convex for a.e. $x$. Furthermore ...
0
votes
1answer
47 views

Prove $mx+b$ is continuous at any point in $\mathbb{R}$

I need to prove, $mx+b$ is continuous at any point in $\mathbb{R}$ Now, as I have thought of there's 2 possible cases: 1) $m = 0$ 2) $m \neq 0$ So for case #2, $m < 0 \vee m > 0$ , and we ...
0
votes
0answers
101 views

Verification of $\epsilon$-$\delta_\epsilon$ definition of continuity proof

Show that the function $$ f(x) = \begin{cases} \frac{1}{q}, & \text{if $x = \pm \frac{p}{q}$ in the lowest terms with $p, q \in N$} \\ 0, & \text{if $x \in R - Q$} \end{cases}$$ ...
1
vote
1answer
45 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
34 views

Introduction to Analysis: Bisections

I feel as though I may be over thinking this problem, at the same time I feel like I may be under thinking it. Use a bisection argument to prove that if $f:[a,b] \rightarrow \mathcal{R}$ is ...
2
votes
1answer
137 views

How to show that time-dependent norm is continuous (please verify my proof)

For each $t \in [0,T]$, let $H_t$ be a Hilbert space. Suppose for each $t$, the operator $T_t:H_0 \to H_t$ is a linear homeomorphism with inverse $T_{-t}:H_t \to H_0$ also linear homeomorphism. ...
1
vote
1answer
66 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) ...
1
vote
1answer
44 views

Introduction to Analysis: Locally and Actually Constant

I was given this problem for homework. I more a less understand it. I just need to somehow finalize my ideas. The problem reads: Prove that a function which is locally constant on $[0,1)$ is ...
0
votes
1answer
102 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 ...
1
vote
2answers
211 views

Is this epsilon-delta proof correct?

Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$ $$f(x)=\begin{cases}x,\ x\in\mathbb{Q} \\ -x,\ x \notin \mathbb{Q}.\end{cases}$$ I'm trying to prove that for all $a \neq 0$, $\lim_{x \to ...
0
votes
1answer
354 views

Show that this real function is Lipschitz continuous

I have this excersize: Let $I$ be an interval in $\Bbb R$ and $f:I\to \Bbb R$ a differentiable function such that $sup_{x\in I}|f'(x)|<\infty$. Show that $f$ is Lipschitz continuous. Well, I ...
0
votes
1answer
116 views

Proof that $\max(x_1,x_2)$ is continuous

I haven't done a proof in years and I've become a little stuck on this, I'd appreciate it if somebody could tell me if I've approached the problem correctly... Question: Prove that the following ...
3
votes
1answer
108 views

$O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb R).$

Let $M(n,\mathbb R)$ be endowed with the norm $(a_{ij})_{n\times n}\mapsto\sqrt{\sum_{i,j}|a_{ij}|^2}.$ Then the set $O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb ...
1
vote
1answer
64 views

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous.

Show that for $n\in\mathbb N$ the mapping $f:M(n,\mathbb R)\to M(n,\mathbb R):A\mapsto A^n$ is continuous. ($M(n,\mathbb R)$ is identified with $\mathbb R^{n^2}$ as a normed liner space.) ...