0
votes
2answers
40 views

Example of differentiable function with $f'(s_{n})=0$ but $f'(0)>0$

Ex: Give an example of a differential function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $0$ is a limit (accumulation) point of a sequence of critic points ($f'(x)=0$) and however $f'(0)>0$ ...
-1
votes
1answer
37 views

Finding a function which is onto, monotone and not one-one [closed]

Does there exist a function $f:[0,1] \to [0,1]$ which is onto, monotone and not one-one?
0
votes
1answer
84 views

How to construct a smooth curve whose range is dense in $\mathbb R^2$?

How to construct a smooth curve $f: \mathbb R \to \mathbb R^2$ whose range is dense in $\mathbb R^2$? Space-filling curves are well-known, but they cannot be smooth. The image of a smooth ...
3
votes
1answer
57 views

A derivative which is not Lebesgue integrable on any interval?

If $f=x^2\sin(x^{-2})$, then $f'$ exists everywhere (including $x=0$) but $f'$ is not Lebesgue integrable on $[0,1]$ (precisely because of the singularity at $x=0).$ I'm trying to find a function $f$ ...
0
votes
1answer
33 views
1
vote
1answer
35 views

A Real valued function which is discontinuous **only** on a given specific set.

Let $\mathbb{L}=\{x_n \ |\ n=1,2,3 \dots\}$ be a countable subset of $\mathbb{R}$. My aim is to construct a real valued function $f$ on $\mathbb{R}$ such that $f$ is discontinuous at every point ...
1
vote
2answers
55 views

Can the derivative of an absolutely continuous real function have a simple discontinuity?

If $f'$ exists everywhere, then we know that it cannot have any simple discontinuities. But in this case we only know that $f'$ exists a.e. (since $f$ is absolutely continuous). More specifically, ...
1
vote
2answers
56 views

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges?

Is it true that if $x_n$ converges and $y_n$ is bounded, then $x_ny_n$ converges? $x_n$ is said to be bounded if and only if it is bounded both above and below. I believe this to be false. My ...
0
votes
1answer
24 views

A function with only a partial derivative not Hölder-continuous

I'm looking for a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$ 1. $x\mapsto u(t,x)$ is $C^{2,\alpha}$; 2. $t\mapsto u(t,x)$ is $C^{1,\alpha}$; 3. $t\mapsto ...
0
votes
1answer
41 views

Question on Taylor's theorem

Taylor's Theorem Let $\{c_n\}$ be a sequence of complex numbers. Let $R$ be the radius of convergence of $\sum c_n z^n$. Let $|b|<R$ and $f(x)=\sum c_n z^n$ on the disk $B(0,R)$. ...
0
votes
2answers
34 views

Can analytic function be represented as a taylor series?

Let $D$ be open in $\mathbb{K}$. Let $f:D\rightarrow \mathbb{K}$ be an analytic function. Then, $f$ is infinitely differentiable and $\forall x_0\in D$, there exists a neighborhood $N$ of $x_0$ such ...
3
votes
2answers
55 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
4
votes
3answers
109 views

Is every such function convex or concave?

I was pondering convex functions today, and the following questions naturally posed themselves. Call a function $f : \mathbb{R} \rightarrow \mathbb{R}$ $2$-limited iff for all $m,c \in \mathbb{R},$ ...
2
votes
1answer
42 views

Discontinuous function whose restriction on closed sets is continuous

Let $X$ a metric space, $\{U_i\}$ a collection of non-empty closed sets whose union is all of $X$. Give an example of a function $f:X\rightarrow \mathbb{R}$ such that the restriction $f|_{U_i}$ is ...
0
votes
1answer
49 views

Are there standard parameters for the Weierstrass nowhere differentiable function?

On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$ where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$ Since it seems like, ...
11
votes
2answers
219 views

Do differentiable functions preserve measure zero sets? Measurable sets?

Consider Lebesgue measure on $\mathbb{R}$ and let $f:\mathbb{R}\to\mathbb{R}$ be differentiable. Does $f$ necessarily preserve measure zero sets? Does $f$ necessarily preserve measurable sets? ...
0
votes
1answer
46 views

Example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable?

Can anyone come up with an example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable? Thanks.
2
votes
2answers
53 views

Convergence of integrals but $\int_a^b|f_n(x)-f(x)|dx$ does not converge to $0$

Can I have an example of a sequence of continuous functions $(f_n)_n$ and a continuous function $f:[a,b]\to \mathbb{R}$ such that $$\int_a^bf_n(x)dx\to\int_a^bf(x)dx,$$ when $n\to+\infty$, but ...
0
votes
1answer
51 views

A counter-example to Dini's Theorem (after removing a hypothesis)

Recall Dini's Theorem: Let $K$ be a compact metric space. Let $f: K\to\mathbb{R}$ be a continuous function and $f_{n}: K\to\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of continuous functions. ...
2
votes
1answer
53 views

Fundamental theorem of calculus necessary assumption

The fundamental theorem of calculus is stated as follows: Let $f$ and $F$ be real-valued functions defined on a closed interval $[a, b]$ such that the derivative of $F$ is $f$. That is, $f$ and $F$ ...
3
votes
1answer
39 views

Example of a function f that is Generalized Riemann Integrable, but its square is NOT Generalized Riemann Integrable.

I am reading a section about Generalized Riemann Integral (Kurzweil-Henstock), and there was a problem on that section to provide an example of a function $f$ on $[0,1]$ that is Generalized Riemann ...
2
votes
1answer
85 views

Unbounded variation but differentiable everywhere

A function with bounded variation is differentiable almost everywhere. There are also functions with unbounded variation that are differentiable almost everywhere (e.g. take ...
2
votes
1answer
99 views

Equicontinuity of a pointwise convergent sequence of monotone functions with continuous limit

I was looking at this question, and trying to come up with a counterexample. After thinking about it, I thought the following might be true: Claim: let $\{f_n\}$ be a sequence of continuous, ...
4
votes
1answer
92 views

Some thinking about the Dini' s Theorem.

In the Dini's Theorem, On the compact set $K$,if $f_n$ is a sequence of monotone increasing or decreasing continuous functions, i,e $f_n(x)\leq(\geq) f_{n+1}(x)$ for all $n$ and all $x$, converges ...
1
vote
2answers
60 views

Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...
1
vote
1answer
120 views

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ and $\sum \sqrt{a_n}$ always convergent? [closed]

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ and $\sum \sqrt{a_n}$ and $\sum \sqrt{a_na_{n+1}}$ always convergent? Either prove it or give a counter example. I am thinking ...
0
votes
1answer
44 views

WHat would be an example of this function?

WHat would be an example of a Lebesgue integrable function $f:\mathbb{R}\rightarrow [0,\infty)$ satisfying: $f$ is continuous. $\limsup_{x\to \infty} f(x)=\infty$
1
vote
0answers
19 views

F sigma subsets of the real line

I need an example of a nowhere dense $\mathcal F_\sigma$ subset of the real line that is not a countable union of perfects sets and not a countable union of pairwise disjoint closed sets.
6
votes
2answers
92 views

A function such that $f'(x)>0$, but not strictly monotone increasing.

This is Exercise 10.3.5 from Analysis Vol.1 by Terence Tao. Give an example of a subset $X \subset \mathbb{R}$ and a function $f: X \to \mathbb{R}$ which is differentiable on $X$, is such that ...
1
vote
1answer
65 views

Continuity conditions for multivariate functions.

Is the following true ? A proof or counter-example or reference would be nice. A function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ is continous at $(0,0)$ if and only if if for all $a, b$, the limits ...
2
votes
1answer
47 views

$\mathcal{C}^\infty$ strictly monotone function $\lim f(x) = 0$, but $\lim f^\prime(x) \ne 0$

Does there exist a strictly monotone function $f\colon \Bbb R\to\Bbb R$ that is $\mathcal{C}^\infty$, and $\lim\limits_{x\to+\infty} f(x) = 0$, but $$\lim_{x\to+\infty} f^\prime(x) $$ and ...
1
vote
1answer
52 views

Counterexample to Marcinkiewicz

I have a version of Marcinkiewicz: Let $(X,\mu)$ and $(Y,\nu)$ be measure spaces and let $1<p_1 \leq \infty$. Suppose that $T$ is a mapping from $L^1(X,\mu) + L^{p_1}(X,\mu)$ to $\mu$- measurable ...
3
votes
1answer
87 views

If partial sums of derivatives diverge then derivative does not exist?

Let $f_n :[a,b]\to \mathbb R$ be differentiable and such that $\sum_n f_n'(x)$ diverges for all $x \in [a,b]$. Let $\sum_n f_n $ converge uniformly on $[a,b]$ to $f$. Does it follow that $f'$ does ...
0
votes
0answers
42 views

Example of sequence for which ratio test is unbounded

I did the following exercise (about ratio test): Show that if $\left | {a_{n+1} \over a_n} \right |$ is unbounded then the original series $\sum_n a_n x^n$ converges only when $x=0$. At first it ...
1
vote
1answer
51 views

Some examples of local and nonlocal properties

Today I learned that continuity at a point is a local property. Concretely, if $f: \mathbb R \to \mathbb R$ is continuous on $[-K,K]$ for all $K \in \mathbb R$ then $f$ is continuous on $ \mathbb ...
3
votes
2answers
91 views

A real continuous periodic function with two incommensurate periods is constant.

I think I have a proof for the statement, but I can't think of a counter-example when $f: \mathbb{R} \to \mathbb{R}$ is not continous. Here's the problem: Let $f: \mathbb{R} \to \mathbb{R}$ be a ...
1
vote
2answers
51 views

Is a bounded continuous function defined on $\Bbb R$ differentiable?

Is a bounded continuous function defined on $\Bbb R$ differentiable? Why so? The query is fueled by the following question: Let $f : \Bbb R \rightarrow \Bbb R$ be a bounded continuous function. ...
1
vote
0answers
34 views

WHat is an example of a countable subset of $[0,1]$ whose Jordan content is 1 and Lebesgue measure is 0?

E.Stein Real analysis p.41 Exhibit a countable subset $E\subset [0,1]$ such that $J(E)=1$ while $m*(E)=0$. Here, $m*$ denotes the outer Lebesgue measure and $J$ denotes the Jordan content. ...
6
votes
3answers
60 views

Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate.

Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$ Is there an example of this ...
0
votes
1answer
24 views

Function differentiable at one point and nowhere else continuous.

Is it possible to construct such a function? Just wondering. Specifically, I am thinking of $f:\mathbb{R}\to\mathbb{R}$ such that $f'(0)$ exists and $f$ is discontinuous for all ...
0
votes
1answer
62 views

Example of a Riemann integrable function which is not a simple function

I'm looking for an example of a Riemann integrable function which isn't simple? I know that all simple functions $f: I \rightarrow E $ ( where $I \subset \mathbb{R}$ is an interval and $E$ is a ...
12
votes
6answers
289 views

Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist?
1
vote
2answers
129 views

Sigma-algebra requirement 3, closed under countable unions.

The requirement for sigma-algebra is that. It contains the empty set. If A is in the sigma-algebra, then the complement of A is there. 3. It is closed under countable unions. My question relates ...
0
votes
0answers
61 views

Completeness is not preserved under homeomorphism

I am trying to give a counterexample which will make it clear that homeomorphism does not preserve completeness. I have an easy one in mind ($(0,1)$ and $\mathbb{R}$) but I have just thought that ...
2
votes
1answer
58 views

A question on Fourier Transform

Is there a function which is not absolutely integrable but which has a continuous fourier transform? I know that if a function is absolutely integrable then the fourier transform is continuous but I ...
3
votes
1answer
133 views

Does there exist a nowhere differentiable, everywhere continous, monotone somewhere function?

Is there a nowhere differentiable but continuous everywhere function which is monotone in some small interval however small it is? Until now I have seen only the Weierstrass function and it seems to ...
1
vote
1answer
80 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
3answers
154 views

An example of an ordered, uncountable set in $\Bbb R$?

Is there an example of an ordered, uncountable set in $\Bbb R$? My Calculus professor, who likes to keep things simple, defined a sequence in $\Bbb R$ as an "ordered and infinite list of real ...
1
vote
1answer
70 views

Intuition and counterexamples for higher-order derivative test

In the higher-order test we keep differentiating a function till we find the n'th derivative (n being even) to be greater than or less than zero thereby identifying it as a minimum or maximum. My two ...
3
votes
1answer
149 views

A calculus counterexample!

Give me an example of two Riemann-integrable functions $f,g:[0,1]\to[0,1]$ such that $g\circ f$ isn't integrable! I already know the following example: $$f(x)=\begin{cases} 0, & \text{if ...