Tagged Questions
10
votes
2answers
188 views
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Let $f :\mathbb{R}→ \mathbb{R}$ be a function such that $f^2$ and $f^3$ are differentiable. Is $f$ differentiable?
Similarly, let $f :\mathbb{C}→ \mathbb{C}$ be a function such that $f^2$ and $f^3$ ...
5
votes
3answers
51 views
Open, closed and continuous
I have some troubles to understanding something:
We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example
...
0
votes
1answer
33 views
A particular weak subadditivity
Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, consider the following property.
For all $(x^1, ..., x^n) \in \left(\mathbb{R}^n \right)^n$ such that $f(x^i) \geq 0$ $\forall i \in [1,n]$, ...
5
votes
2answers
75 views
Continuous function that is only differentiable on irrationals
Can you help me finding a function $f : \mathbb{R} \rightarrow \mathbb{R}$ that is continuous in $\mathbb{R}$ and differentiable at $x$ iff $x \notin \mathbb{Q}$ ?
Thank you very much !
0
votes
0answers
36 views
Help me prove the supremum property.
Let $A$ and $B$ be nonempty sets and $f$ be a function from any nonempty set $S$ to subset of real number. Prove that
$$\sup_{x \in A} \{ \min \{ \sup_{y \in B} \{ \min \{ f(y) \} \}, f(x) \} \}= ...
1
vote
1answer
50 views
What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?
Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$.
a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $
b) $D = \{y \in Y : ...
4
votes
3answers
88 views
Looking for a differentiable function which behaves somewhat like $\min(x,1)$
Is there a differentiable function $f : [0,2] \rightarrow [0,1]$ such that $f(x) = 0$ iff $x=0$ and $f(x) = 1$ iff $x \in [1,2]$? What about $n$ times differentiable for any $n$, or infinitely ...
1
vote
1answer
52 views
Intermediate Value Property and Discontinuous Functions
This is a general question to which I need help finding a concrete example so that I may understand the concept/strategy better, and any help will be greatly appreciated.
If given a function $F$ that ...
1
vote
1answer
25 views
Approximating Lipschitz funtion by $C^1$ function.
Let $G:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function and $L$ its Lipschitz constant. Suppose that $G(0)=0$. It is known that $G$ is almost everywhere differentiable and $G'\in L^\infty$ with ...
0
votes
1answer
48 views
Additive maps modulo $1$ - what do they look like?
Recently, some convergence problems I have been considering led me to look at additive maps of the torus (which for me is the additive group $T := \mathbb{R}/\mathbb{Z}$).
A map $f:\ T \to T$ is ...
1
vote
2answers
75 views
Show that equation has no solution in $(0,2\pi)$
Hi I want to show that the equation $2=2 \cos(x)+x \sin(x) $ has no solution in $(0,2 \pi)$. Since it is algebraically impossible to solve this equation for $x$ I wanted to ask you whether one of you ...
1
vote
1answer
56 views
nth term test for divergence - help
$$\sum_{n=1}^{\infty} \left(\dfrac{n}{n+1}\right)^n$$
to show that this diverges should I use the $n^{th}$ term test?
So far I have substituted infinity for $n$. Could I use L'hopital's rule to ...
0
votes
1answer
44 views
Supremum and infimum for this set - help needed
what is the set S bounded by? how do I do these questions? can someone please show me an exemplar solution that I could follow - thank you
0
votes
0answers
45 views
mean value theorem question!
Let a function $f$ be a contraction function in $[-a,a]$. Derive conditions on $f$ such that $|f(x)|\leq|x|$ for all $x\in[-a,a]$.
I need to answer this question using the mean value theorem but ...
0
votes
1answer
37 views
A smooth or analytic function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
In A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer, I asked about continuous function. This time, I would like to ask the ...
0
votes
1answer
67 views
For which values of $a$ is $f(x) = \cos(ax)$ a contraction mapping?
I am studying contraction mapping and got stuck on this question:
Consider the following function:
$$f(x)=\cos(ax),\ a,x \in \mathbb{R}.$$
Find values of $a$ for which $f$ is a ...
0
votes
2answers
44 views
A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...
0
votes
3answers
46 views
Domain or range?
I am doing some questions on contraction mapping and the first part of the question is:
I am not sure what the question is asking for and how to do it.
Any help is hugely appreciated! thank you
0
votes
3answers
38 views
Can there be a non-polynomial continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has multiple zero-valued points?
1) Is there any non-polynomial function from $\mathbb{R}$ to $\mathbb{R}$ that has more than one zero-valued point in domain?
2) Is there any non-polynomial function from $\mathbb{R}$ to $\mathbb{R}$ ...
1
vote
2answers
37 views
Looking for function with specific properties
I need a function $f$ that is arbitrarly times differentiable and which has integral
$$\int _a^b f(x) dx $$ strictly positive (where $a$ and $b$ are fixed), and for all derivatives, we have ...
1
vote
2answers
76 views
non continuous ivt problem
solving for an ivt for a non-continuous function
We take $f : [0,1] \to [0,1]$ a non-increasing function, such that $f(x)≥ f(y)$ whenever $x≤ y$;
and we want to prove that there exists $c\in [0,1]$ ...
2
votes
1answer
72 views
Show that some $C^\infty$ real function is analytic
Consider the function $f:(a,b) \rightarrow \Bbb R $, which is $C^\infty$ class. It is required to show that if the function $f$ satisfies: $\forall_{n\in\Bbb N_+} \forall_{x\in(a,b)} f^{(n)}(x) \ge 0 ...
5
votes
3answers
71 views
Continuous function that take irrationals to rationals and vice-versa. [duplicate]
Can someone help me?
How can I prove that there isn't an everywhere continuous function $f:\mathbb R \rightarrow \mathbb R$ that transforms every rational into an irrational and vice-versa?
0
votes
4answers
113 views
Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, … $ but is continuous at every other point
(a) Find a function $f:\Bbb R \to \Bbb R$ which is discontinuous at $1,\frac 12,\frac 13, ... $ but is continuous at every other point.
(b) Find a function $f:\Bbb R \to \Bbb R$ which is ...
1
vote
1answer
58 views
Derivative of a little-o remainder
If we have a $\phi: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$, $\phi = \phi(t, \mathbf{q},\alpha)$ one-parameter group of infinitesimal transformation which is $\mathcal{C}^2$ ...
1
vote
2answers
51 views
product rule for matrix functions?
Given a real rectangular matrix $X$, and two scalar-valued matrix functions, $f(X)$ and $g(X)$, does the product rule for differentiation of a product of scalar valued functions, hold when ...
3
votes
5answers
121 views
Sequence of continuous functions which converges to a continuous limit
Any help with this: construct a sequence of continuous functions defined on $ [0,1] $ which converges pointwise but not uniformly to a continuous limit ?
Thank you.
2
votes
1answer
63 views
If $f '(2) = 0$ and $f ''(2) = 4$, what can you say about $f$?
I was doing very well in Calculus up until this point. I realize that concavity and $f'$ and $f''$ require one to really visualize what is happening with a function, but can someone please help me to ...
0
votes
0answers
68 views
Can a function with finite discontinuities be nicely approximated by a continuous function?
1) Can every discrete function be approximated by some continuous function with regards to domain defined?? (By discrete function, I mean: there is countably infinite number in domain, and for each ...
3
votes
1answer
107 views
Creating a bijection from $(a,b)$ to $\mathbb R$ that is visually compelling
Given this "proof without words" from MO
I am trying to:
a) find a function that behaves like the function shown, on an open interval - say $(-1/2,1/2)$
b) find some intuition for why/how this ...
7
votes
3answers
102 views
Finite at every point but unbounded on every interval
Is is possible that a function $f$ is finite at every point but unbounded on every interval?
What if f is measurable?
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 ...
4
votes
4answers
161 views
Show that function is strictly monotone increasing
I want to show that $$ f(x)=\dfrac{x-\sin(x)}{1-\cos(x)} $$ is strictly increasing in $(0,2 \pi) $. Unforunately, this is not that easy for me , as the derivative is not very manageable and ...
0
votes
1answer
71 views
Upper and lower bound of $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$
Let define $f(x)=(\tan x)^{\sin 2x}$ for $x \in (0, \frac{\pi}{2})$
Please help me prove, that $f$ reaches its lower bound in only one point $x_1$ and reaches its upper bound $x_2$ also in only one ...
2
votes
1answer
31 views
comparing two functions
I am working on some problem and got stuck at one point:
I need to show that if $f(t)=(\log t)^2$ and $g(t)=\frac{(\log \sqrt{x})^2}{\sqrt{x}-1}(t-1)$, then $f\ \ge g\ $ for any $1\le t \le ...
0
votes
1answer
49 views
Monotone function, inverse image of an interval
Could you tell me how to prove that if $f$ is monotone,the inverse image of an interval is an interval?
Does it suffice to say that $f^{-1}(a,b) = (f^{-1} (a), f^{-1}(b))$ ?
0
votes
1answer
100 views
Question about a counterexample related to the mean value theorem for integrals
Let $g(x) = x$ on the interval $[ 1, 3]$. Can you find a function $f (x)$ such that no value between the minimum and maximum of $f (x$) satisfies
$$
\int_{a}^{b}f(x)g(x) dx \,=\, ...
3
votes
1answer
52 views
Non-differentiability in $\mathbb R\setminus\mathbb Q$ of the modification of the Thomae's function
Here is the problem I'm struggling with:
Where is the following function continuous, differentiable, continuously differentiable?
$$f(x) =
\begin{cases}
q^{-2} & \text{if $x=\frac{p}{q}$ ...
1
vote
2answers
31 views
Multiplicities of conjugate roots
If a real polynomial of degree n has a complex root, then it is clear that its conjugate is also a root. But how to verify that the multiplicities of the conjugated roots are equal?
0
votes
3answers
51 views
Is the map $f: \mathbb{U}^2 \setminus \{(1,0)\} \to \mathbb{R}$ surjective?
Define $f: \mathbb{U}^2\setminus \{(1,0)\} \to \mathbb{R}$ by
$f(x,y) = \dfrac{y}{2(x-1)}$. Is this map surjective?
2
votes
3answers
68 views
If a bounded function $f:\Bbb R\to \Bbb R $ and $\left|\,f(x)-f(y)\right|<\left|x-y\right|$ for $x\ne y$, then there is an $a$ s.t. $f(a)=a$.
If a bounded function $f:\Bbb R\to \Bbb R $ and $\left|\,f(x)-f(y)\right|<\left|x-y\right|$ for $x\ne y$, then there is an $a$ s.t. $f(a)=a$.
What I know is $f$ should be uniformly ...
2
votes
1answer
80 views
Check convergence of $f_{n}(x)=x^{n}-x^{2n}=x^{n}(1-x^{n})$
Check convergence of
$f_{n}(x)=x^{n}-x^{2n}$ where $x\in(0,1)$
Please verify my answer, I'm not sure I'm doing it correctly. Thanks in advance!
2
votes
1answer
115 views
Continuous function proof by definition
Prove that if $f$ is defined for $x\ge 0$ by $f(x)=\sqrt x$, then $f$
is continuous at every point of its domain.
Definition of a continuous function is:
Let $A\subseteq\mathbb{R}$ and let ...
0
votes
1answer
20 views
does $f^{-1}(y)=\bar f^{-1}(y-f(x_0))$?
Define a function $\bar f(x)=f(x)-f(x_0), f:R\to R$ i want to ask does $f^{-1}(y)=\bar f^{-1}(y-f(x_0))$? I ask this question as i am not sure if there is a typo in the notes as $f^{-1}(y)=\bar ...
2
votes
1answer
72 views
Can the product of two monotone functions have more than one turning point?
If we have 2 monotone functions $f$ and $g$ non zero, is it possible that $fg$ has more than one turning point. We can assume wlog that $f$ is increasing and $g$ is decreasing.
$\frac{1}{x}e^x$ is an ...
1
vote
2answers
51 views
Showing that a modifying function which is continuous at 0 is uniformly continuous
This is the definition of a modifying function I've got to work with:
In this problem, a function $\phi :[0,\infty)\rightarrow [0,\infty)$ is called a modifying function if
(a) $\phi ...
0
votes
2answers
128 views
find the supremum and/or infimum of this set
I am struggling to find the supremum and/or infimum of this set:
$$ S = \biggl\{ \frac{x^2+y^2}{xy} \;\bigg|\; x>0, y>0 \biggr\}$$
any kind of help will really be appreciated. Thank you
1
vote
1answer
45 views
Showing that a function is a modifying function (how to prove subadditivity)
This is the definition of a modifying function I've got to work with:
In this problem, a function $\phi :[0,\infty)\rightarrow [0,\infty)$ is called a modifying function if
(a) $\phi ...
3
votes
1answer
75 views
Application of Banach fixed-point theorem
I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satifies $\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$ for $t\in[0,1]$.
The first thing I do is to define a function ...
3
votes
1answer
61 views
If $f'(t) = g'(t)$ then $f(t) = g(t) + k$ for all $t\in\mathbb R$
Let $f,g: I \to\mathbb R$ for some interval $I\subset\mathbb R$. Then $f'(t) = g'(t)$ for all $t\in I$ if and only if there exists $k\in\mathbb R$ such that $f(t) = g(t)+k$.
Necessary condition ...
