Tagged Questions
0
votes
0answers
16 views
Unique continuos linear function given a continuous function from a dense space in X to Y (Y is a Banach Space).
Let $X$ be a normed space, let $Y$ be a Banach Space, let $D\subseteq X$ be a dense linear subspace of $X$ and let $L:D\rightarrow Y$ be a continuous linear function. Then there is a unique continuous ...
0
votes
1answer
38 views
Is this function continuous on $[0, 250]$? Is it differentiable?
Consider
$$
I(X)=\begin{cases}
15X & 0≤X≤150\\
22.5X-0.05X^2 & 150<X≤250
\end{cases}
$$
Is $I(x)$ continuous on $[0, 250]$?
Is $I(x)$ differentiable on $[0, 250]$?
THANKS!!
2
votes
1answer
25 views
Differentiability of first derivative of a function
If a function $f$ is differentiable on domain $D$ and $f'$ is increasing on $D$, is $f'$ necessarily continuous on $D$? Is $f'$ necessarily differentiable on $D$? Counterexamples?
From Darboux ...
0
votes
2answers
42 views
Let $f:\Bbb R^2→\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$
Let $$f:\Bbb R^2\to\Bbb R:(0,0)\mapsto 0 \quad (x,y)\mapsto\frac{x^2y^2}{x^4+y^4}$$
i) Is $f$ continuous at $(0,0)$?
ii) Is $f$ differentiable at $(0,0)$?
I can prove that $f$ is ...
1
vote
2answers
42 views
Suppose that $a,c$ are real numbers ,$c>0$ and $f$ is defined on $[0,1]$ by…
Suppose that $a,c$ are real numbers , $c>0$ and $f$ is defined on $[0,1]$ by $f(x)=x^a \sin (|x|^{-c})$ if $x \neq 0$ and $f(x)=0$ if $x=0.$ Then I have to show that
1.$f$ is continuous iff ...
0
votes
0answers
77 views
Continuous, differentiable, continuously differentiable
I came across the following problem:
Let $\alpha \in \mathbb R$. Where is the function continuous, differentiable, continuously differentiable?
$$f(x) =
\begin{cases}
x|x|^\alpha & ...
0
votes
1answer
31 views
Show that the function is not continuous in point $x_{0}=0$
I should show that $f':\mathbb{R}\rightarrow\mathbb{R}$ is not continuous at the point $x=0$, where $$ x \mapsto \begin{cases} 2x \cos{\left(\frac{1}{x}\right)}+\sin{\left(\frac{1}{x}\right)} & ...
1
vote
1answer
74 views
Alternative sufficient conditions for differentiability of two-variable functions?
Does anyone know of a counterexample (or proof) for the following?
Suppose $f: D \to \mathbb{R}, D \subset \mathbb{R}^2$ is continuous at $(a,b)$ and its directional derivatives are linear in the ...
1
vote
3answers
52 views
Why is it clear from this formulation that f is continuous wherever it is holomorphic?
Hi I am new on here so not sure if this is right place to post but quick and presumably easy question:
So holomorphic at a point $z_0 \in \Omega$ is defined as the limit as $h\rightarrow 0$ of
...
0
votes
1answer
56 views
Continuity of a function, Differentiable function
The following function is given:
$$f:\mathbb{R}\rightarrow \mathbb{R}, \ x\rightarrow \begin{cases} x^2\cos{\left(\frac{1}{x}\right)} & \text{for } x \neq 0\\ 0& \text{for } x =0\end{cases}$$
...
4
votes
1answer
93 views
Why doesn't Dirichlet function have a derivative in X=0
$\newcommand{\dirichlet}{\mathop{\rm dirichlet}\nolimits}$
I'm trying to find two examples for the following criterias:
A method that is continuous in exactly one point but doesn't have a derivative ...
2
votes
1answer
54 views
Suppose the function $f:\mathbb R \rightarrow \mathbb R$ has left and right derivatives at $0$.
I have been trying to solve the following problem:
Suppose the function $f:\mathbb R \rightarrow \mathbb R$ has left and right derivatives at $0$.Then at $x=0$, which of the following options is ...
1
vote
1answer
30 views
Showing that a certain function is $C^1$
For an exercise in my analysis course, I have to show that the function
$$\newcommand{\sgn}{\operatorname{sgn}}f: (x,y) \mapsto \begin{cases} \frac{(x \sin y)^2}{|x|+|y|},&(x,y) \neq (0,0) \\
...
1
vote
2answers
109 views
Derivative based on continuity
I have a question about whether I am even close to correct.
Let $\mathbb{I}$ and $\mathbb{J}$ be open intervals, and the functions $f:\mathbb{I} \to R$ and $h:\mathbb{J}\to R$ have the property that ...
3
votes
2answers
142 views
Using LDCT to show a function is continuous and differentiable
We have the following test prep question, for a measure theory course:
$\forall s\geq 0$, define $$F(s)=\int_0^\infty \frac{\sin(x)}{x}e^{-sx}\ dx.$$
a) Show that, for $s>0$, $F$ is ...
1
vote
2answers
201 views
The limit of the sum is the sum of the limits
I was wondering why the statement in the title is true only if the functions we are dealing with are continuous.
Here's the context (perhaps not required): http://i.imgur.com/xbAha.png (The upper ...
0
votes
3answers
148 views
Show that this function is not increasing on any interval containing $0$:
$$f(x) = \begin{cases}x + 2x^2\sin\left(\frac1x\right),& x\ne 0\\0,& x = 0\;.\end{cases}$$
I am having a tough time answering this question in a rigorous mathematical way, here is what I have ...
8
votes
8answers
1k views
Continuous versus differentiable
A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it.
Until today, I thought these were merely two equivalent definitions of the same ...
