5
votes
2answers
98 views

Give an example of a function $f$ satisfying $\lim_{x\to 0}(f(x)f(2x))=0$,but $\lim_{x\to 0}f(x)$ does not exists

Question: Give an example of a function $f$ satisfying the condition $$\lim_{x\to 0}(f(x)f(2x))=0$$ and such that $$\lim_{x\to 0}f(x)$$ does not exists. I think this question have many example. But ...
1
vote
3answers
132 views

A counter-example to differential function but not twice differential

Find a function $f$ that is differentiable, but not twice differentiable and which does not belong to the following type: $$f(x) = \begin{cases} x^\alpha \sin(x^{\beta}) & x \neq 0 \\ 0 & ...
6
votes
2answers
71 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 ...
0
votes
0answers
22 views

is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
2
votes
1answer
41 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
43 views

L'Hospital's rule vs Taylor series

One classical application of Taylor expansions is to obtain polynomial equivalents of complicated functions and use them to compute limits. For example, with Landau notations, we have ...
6
votes
1answer
172 views

A counterexample

Let $f:\mathbb{R}\to(0,\infty)$ a locally integrable function. I want to compare these two conditions $$\limsup_{r\to + \infty}\frac{r}{\int_{-r}^r f(x)dx}<+\infty. \tag{1}\label{1}$$ and ...
0
votes
0answers
12 views

smooth function whose (n+1)th derivative is defined only on a propersubset of the domain of the nth, and the radius contract to 0

So I'm basically wondering if there exists such a function, whose (n+1)th derivative is defined only on a proper subset of the domain where the nth derivative is defined, and with the property that ...
3
votes
2answers
63 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
49 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. ...
0
votes
2answers
43 views

If $\lim \limits_{x \to \infty}f(x) = L \neq 0$ must it be that $\lim_{x \to \infty} f(x) \sin x $ does not exist?

I got this question: Let $f\colon \mathbb{R} \to \mathbb{R}$ be a function that satisfies $\lim \limits_{x \to \infty}f(x) = L \neq 0$, Must it be that $\lim_{x \to \infty} f(x) sin x $ does not ...
8
votes
5answers
729 views

Are continuous functions monotonic for very small ranges?

So I am wondering, if we have a continuous function f and we take the range $[c,c+h]$ for $h \to 0$, is the function monotonic in that range?
3
votes
1answer
147 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 ...
8
votes
3answers
241 views

Examples of non-Riemann integrable functions that appear “in nature”?

I am teaching an honours calculus class, and am looking for examples on non-integrable functions that occur somewhere real in mathematics. (The standard example of 1 on $\mathbb{Q}$ and 0 elsewhere ...
7
votes
9answers
363 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...
2
votes
2answers
232 views

Show a convergent series $\sum a_n$, but $\sum a_n^p$ is not convergent

$p>1$ is a integer, Show a convergent series $\sum\limits_{n=1}^\infty a_n$, $a_n\in\Bbb R$, such that the series $$\sum_{n=1}^\infty a_n^p$$ is divergent p.s. If $p>1$ is not an integer ...
2
votes
1answer
203 views

Existence and uniqueness of limit of inverse function

Let $f:(a,b) \rightarrow \mathbb{R}$ be a one to one function. If $x_0$ is a point of the open interval $(a,b)$ such that $\lim_{x \rightarrow x_0} f(x) = l$, is it necessary that $\lim_{x \rightarrow ...
3
votes
1answer
134 views

Counterexample to second derivative test when f''(x) is not continuously differentiable

When looking over true/false questions on previous midterms, one of my conscientious students said: "If f is defined on an open interval containing c, f'(c)=0, and f''(c)>0, then c is a local min of ...
3
votes
1answer
112 views

If $f(x)$ is positive and decreasing, can $xf(x)$ have more than one maxima?

Assume that $f(x)$ is positive and decreasing on $[0,1]$ with $f(0)=1$ and $f(1)=0$. We see that $xf(x)$ is 0 at 0 and 1 and is positive in between so it must have a maxima. Is it possible that ...
4
votes
1answer
61 views

For this periodic continuous $g:\Bbb R\to \Bbb R$, and $f_n(x):=g(x/n)$, does $\{f_n\}_{n=1}^\infty$ converge uniformly?

I can not find a counterexample although I have the feeling it is not true. Let $\ g: \mathbb{ R} \rightarrow \mathbb{R}$ continuous function $ \forall x \in \mathbb{R} \ g(x+1) = g(x)$ $g(0) = 0$ ...
5
votes
1answer
323 views

Two variable limits via paths - are there pathalogical examples?

In the first year calculus course at my university, we do not introduce the $\varepsilon$-$\delta$ definition of a limit. When considering the limit of a function of two variables, we resort to paths. ...
17
votes
6answers
797 views

If $f$ is continuous at $a$, is it continuous in some open interval around $a$?

If $f: \mathbb{R} \to \mathbb{R}$ is continuous at $a$, is it continuous in some open interval around $a$?
1
vote
1answer
52 views

Show that floor function does't satisfy FTC.

The function is $f(x) = \lfloor 1-x^2 \rfloor$. $$f(x) = \left \{ \begin{array}{lr} -3 & : x \in [-2,-\sqrt{3})\\ -2 & : x \in [-\sqrt{3},-\sqrt{2})\\ -1 & : x ...
4
votes
1answer
149 views

Is a $C^\infty $ function local monotone at a minimum?

The following is a modified exercise from an analysis 1 book. Is there a function $f:\mathbb{R} \rightarrow \mathbb{R}$ with: i) $f$ has in $0$ a strict local and global minimum. ii) $f\in C^\infty$ ...
4
votes
2answers
207 views

Existence of an infinitely differentiable function $ f $ with $ {f^{(n)}}(0) = 0 $ for all $ n \in \mathbb{N} $.

How can one show that there exists an infinitely differentiable function $ f: \mathbb{R} \to \mathbb{R} $ such that $ {f^{(n)}}(0) = 0 $ but $ f^{(n)} \not\equiv 0 $ for all $ n \in \mathbb{N} $?
5
votes
3answers
309 views

If $|f(x)|$ is a differentiable function, then $f(x)$ is also?

If $|f(x)|$ is a differentiable function, then $f(x)$ is also a differentiable function. Why is this wrong? Can you find a counterexample please? It seems like a true sentence.
7
votes
3answers
1k views

Non-physical Jounce Examples in Nature

What are some good examples of jounce, the fourth derivative of position, in the non-physics arena? The reason I ask is that A) it's already difficult for a lay to visualize it in the physical arena ...
6
votes
2answers
420 views

nonstandard example of smooth function which fails to be analytic on $\mathbb{R}$

When I teach second-semester calculus I usually discuss the function $f$ defined by $$ f(x)=e^{-1/x^2} $$ for $x \neq 0$ and $f(0)=0$. Or, almost the same example, $g$ defined by $$ g(x)=e^{-1/x^2} $$ ...
1
vote
1answer
105 views

Is there any complex-valued $C^\infty$ function $f(z)$ coincide with $z^{3/2}$ infinitely many times?

I want a $C^\infty$ function $f:U\rightarrow\mathbb{C}$, where $U\subset\mathbb{C}$ is a neighborhood of $0$, such that there exists a sequence of points $z_n\in U-\{0\}$ and ...
3
votes
2answers
116 views

Sequence convergence and parentheses insertion

find an example for a series $a_{n}$ that satisfies the following: $a_{n}\xrightarrow[n\to\infty]{}0$ ${\displaystyle \sum_{n=1}^{\infty}a_{n}}$ does not converges There is a way to insert ...
12
votes
2answers
663 views

Set of zeroes of the derivative of a pathological function

For a continuous function $f : [0,1] \to {\mathbb R}$, let us set $$ X_f=\lbrace x \in [0,1] \bigg| f'(x)=0 \rbrace $$ (for a $x\not\in X_f$, $f'(x)$ may be a nonzero value or undefined). There ...
0
votes
1answer
452 views

Real-valued function of one variable which is continuous on [a,b] and semi-differentiable on [a,b)?

Is there any real-valued function of one variable which is continuous on [a,b] and right differentiable on [a,b), but not left differentiable at any point?
4
votes
1answer
247 views

An explicit example of a differentiable function taking rational values at rational points but whose derivative is irrational at rational points

Construct an example of a differentiable function such that $$ \forall r \in {\Bbb Q}\quad f(r) \in {\Bbb Q}\text{ but } f'(r) \notin {\Bbb Q} $$ this example is not trivial, in a paper they ...
18
votes
2answers
780 views

“Pseudo-Cauchy” sequences: are they also Cauchy?

I tried to prove something but I could not, I don't know if it's true or not, but I did not found a counterexample. Let $(a_n)$ be a sequence in a general metric space such that for any fixed $k ...