0
votes
1answer
21 views

An example of a twice continuously differentiable and bounded function.

Find an example of a twice continuously differentiable and bounded function $f:\Bbb R \rightarrow\Bbb R$ such that $\lim\limits_{x \rightarrow \infty} f(x)$ exists, but $\lim\limits_{x\rightarrow ...
2
votes
2answers
30 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
0
votes
0answers
46 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
1
vote
3answers
49 views

Is Inverse Function Composition Commutative?

Given $f: \mathbb{R}\to(-1,1)$ is there a theorem that states $f\left(f^{-1}(x)\right) = f^{-1}\left(f(x)\right)$. In example, is $\tanh{\left(\tanh^{-1}{(x^2)}\right)} ...
3
votes
2answers
42 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
1
vote
1answer
34 views

Having derivative in some $x_{0}$ implies having it in $U(x_{0})$ [closed]

Let $f$ be continious function in R and it has a derivative in $x_{0}$. Does it have derivative in some $U(x_{0})$?
0
votes
1answer
124 views

if $\mid f(x)\mid<1$ then $\mid f'(x)\mid<1$

I am wondering, if $\mid f(x)\mid<1$ then $\mid f'(x)\mid<1$. Is this true? I can not find counter example.
1
vote
0answers
29 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
31 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
1
vote
1answer
39 views

Find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$

How can I find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$. I've tried derivating it but didn't reach any result.
0
votes
0answers
37 views

Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
1
vote
2answers
98 views

Finding a root of a function via Rolle's theorem

Consider the function $f(t)=a(1-t)\cos(at)-\sin (at)$, where $a\in\mathbb R$. To show that it has a root in the unit interval I am urged to integrate $f$ and apply Rolle's Theorem. Attempt: $$\int ...
1
vote
2answers
59 views

If a uniquness for all functions exist shouldn't there be uniquness to recursion?

What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example. Let's make a table of values for $^{x}2$ (0,1) (1,2) (2,4) ...
2
votes
1answer
49 views

Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?

I had intended to restrict the image then $f:U \rightarrow f(U) \subset \mathbb{R}^m $ is bijective. Therefore $\dim f(U) = n \leq m$. That's right?
2
votes
1answer
33 views

Election measurable in uniform continuity

Let $f:[0,1]\times [0,1] \rightarrow \mathbb{R}$ borel measurable such that for all $x \in [0,1]$ $f(x,-):[0,1] \rightarrow \mathbb{R}$ is continuous, in particular uniformly continuous. Then there ...
2
votes
1answer
83 views

Show that $e^{-a|x|}$ does not belong to Schwartz space

Let $f : \mathbb R \to \mathbb R$ and $a > 0$ given by $f(x) = e^{-a|x|}$. Show that $f$ is rapidly decreasing and belongs to $L_1(\mathbb R)$, but not to $\mathcal S(\mathbb R)$. I had shown that ...
0
votes
2answers
45 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
4
votes
2answers
66 views

$f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
0
votes
3answers
65 views

Example of a function?

$f$ is a discontinuous and bounded function defined on a closed set $C$. Also there exists a non-discrete closed subset in the image of $f$ such that it's inverse is open. Can you give an example ...
2
votes
0answers
41 views

Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
0
votes
1answer
44 views

Interesting question regarding elementary functions

I had this question at a test for a job interview and since and I didn't solve it. Some time later i still can't figure it out, so any insight is helpful. You need to write a function $f(x)$ such ...
0
votes
1answer
31 views

Help me how to find the limit. In this case something is strange.

A fn(x) is a sequence of function. fn:[0,1]→R defined by fn(x) = n(1-x) × x^n. And I want to know limit of this function as n→∞. In my opinion we can see this like this n(1-x) × x^n. So as n→∞, ...
0
votes
2answers
34 views

Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
0
votes
1answer
60 views

Linearly approximating a curve [closed]

I'm trying to approximate y = x^3 with straight lines. One of the points at the end of two of the linear segments is (2,8). The linear segments are 22.5 units long. What are the coordinates of the ...
0
votes
1answer
28 views

Extension of a holomorphic function in the disc

let $f$ be a continuos function in ${0<|z| \leq r} $ holomorphic in the inner and such that $f(z) $ is real for $|z|=r$. Prove that exist a function $g$ on $\mathbb{C} ^*$ such that $f(z) =g(z) $ ...
1
vote
0answers
26 views

$f(z) = u(x,y) + i\cdot v(x,y)$ holomorphic in a connected open set $D$, such that $a\cdot u(x,y)+b\cdot v(x,y)=c$, is constant

Let $f(z) = u(x,y) + i\cdot v(x,y)$ be a holomorphic function in a connected open set $D$. If $a\cdot u(x,y)+b\cdot v(x,y)=c$ in $D$, where $a,b,c$ are real constants which are not all zero, why ...
1
vote
1answer
23 views

A continuos and holomorphic function on $D^2$ that take pure imaginary values on $S^1$ is costant

Let $D := \{ |z| < 1\}$ and $f : \overline{D} \rightarrow \mathbb{C}$ be a continuos and holomorphic function on $D$ that take pure imaginary values on $\partial D$. Why $f$ is constant? From ...
2
votes
1answer
52 views

An open map from $\mathbb{C} \rightarrow \mathbb{C}$ has open real and imaginary part?

If $f(z) :\mathbb{C} \rightarrow \mathbb{C}$ is an open map such that $f(z) = f_1(z) + if_2(z)$ where $f_1$ and $f_2$ represent respectively his real and imaginary part, we could say that both $f_1$ ...
0
votes
0answers
15 views

Estimate error when using the result of an expanded equation

I would like to know how to deal with the error term in expanded expression. For example consider the function $\displaystyle f(x)=A\text e^{-(x+\lambda)^2}+B\text e^{-(x-2\lambda)^2}\;, $ where ...
2
votes
1answer
113 views

How many extrema has $f(x)$ from $L^2$ if it is not a polynomial?

Is it possible to say how many extrema the $L^2$ function $ f(x)=A\text e ^{-(x+\lambda)^2} - B\text e ^{-x^2} + C\text e ^{-(x-\lambda)^2} $ has on whole $\mathbb R$? Here $A$, $B$, $C$ and ...
6
votes
1answer
114 views

How prove $f(x)$ is monotonous , if $f'(x)=g[f(x)]$

Question: Let $f(x)$ be a derivative, and there exsit $g(x)$ be such that: $$f'(x)=g[f(x)]$$ Show that $f(x)$ is monotonic. This problem is from Xie Hui Min analysis problems book in china ...
0
votes
2answers
47 views

Small question about limit

if i have $\displaystyle \lim_{|u|\rightarrow 0}\frac{f(t,u)-a|u|^{\tau-2}u}{u}=0$ how to prove that $\displaystyle \lim_{|u|\rightarrow 0}\frac{f(t,u)}{|u|^{\tau-2}u}=a$ such that $\tau\in (1,2)$ I ...
1
vote
1answer
47 views

Where to stop a taylor expansion of a function of more than one variable?

I tried to taylor expand a function of three variables $f(x,y,z)$ around $y=0,z=0$ but I don't find a way to decide where to stop the expansion. Usually when one expands a function of one variable ...
1
vote
2answers
42 views

To calculate the limit :

$$\lim_{n\rightarrow\infty}{n^2}(\arctan\frac{a}{n}-\arctan\frac{a}{n+1})$$ I used the formula $\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$,but it just doesn't work. Waiting for your help...
0
votes
1answer
23 views

Contraction map

I have a general question about the properties of contractive/non-contractive maps. Assuming I have a map $F$ which I know is not a contraction. Can it map a ball back to itself, i.e. for some ...
2
votes
1answer
37 views

right derivative of a continuous function

Let $f:(a,b)\longrightarrow \mathbb{R}$ be continuous. Suppose $D_+f(x)=\lim_{h\to 0+}\frac{f(x+h)-f(x)}{h}\geq 0$ for any $x\in (a,b)$. Prove that $f(x_1)\geq f(x_0)$ whenever $x_1\geq x_0$. How to ...
0
votes
2answers
41 views

construct functions such that $f(x)g(x)\gt0 $ and

Does there exist real functions $f, g\in C^1[-1,1]$ such that $$\det\left(\begin{array}{cc}f &g \\ f'&g'\end{array}\right)\equiv0 \qquad \det\left(\begin{array}{cc}\int_{-1}^1f^2\,\mathrm ...
1
vote
1answer
18 views

Show that the following functi0n is bounded

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $f(0)>0$ and $$\lim_{x \to \infty} f(x)= \lim_{x \to -\infty} f(x)=0$$ $(i)$ Show that $f$ is bounded. $(ii)$ Let ...
0
votes
0answers
23 views

double limits and liminf, limsup

A theorem in calculus: Let $f:\mathbb{R}^2\longrightarrow \mathbb{R}$. If $\lim_{(x,y)\to (a,b)}f(x,y)=A$ exists (either finite or $\infty$) and there exists $\epsilon>0$ such that for any ...
0
votes
0answers
8 views

Concerning the fiberwise measure

Given a map from a space $X$ to a space $Y$ and a measure $m$ on $Y$, could someone tell me the precise definition of the fiberwise measure on $M$? I have heard about it but I can't find any ...
1
vote
1answer
19 views

Convergence and uniform convergence of a sequence of functions

Suppose that ($f_n$;$n \geq 1$) is a sequence of functions defined on an interval [a,b].We will say $f_n$ tends to $f$ uniformly on $[a,b]$ as n tends to infinity if for every $\epsilon>0$ there ...
1
vote
1answer
23 views

sequence of analytic functions on an open subset of $\mathbb{C}$ that converges uniformly on compact subsets

Let $U$ be an open subset of $\mathbb{C}$ and let $\{f_n\}$ be a sequence of analytic functions on $U$. Suppose that $(f_n)$ converges uniformly on any compact subsets of $U$ to a function $f$. Let ...
4
votes
1answer
47 views

construct a function

I wonder whether there is a function $f\colon\Bbb R\to\Bbb R$ with the folowing characteristic? for every two real numbers $\alpha,\beta,\alpha\lt\beta$, $$\{f(x):x\in(\alpha,\beta)\}=\Bbb R$$ ...
2
votes
1answer
38 views

Let $f_n: D \rightarrow \mathbb{R}: f_n(x) = g(x)^n, n≥1$. Necessary and sufficient conditions such that $f_n$ converges?

The Assignment: Let $D := [a,b]$ with $a<b$ and $g: D \rightarrow \mathbb{R}$ be continuous. Observe the sequence of functions $f_n: D \rightarrow \mathbb{R}: f_n(x) = g(x)^n, n≥1$. List and ...
1
vote
1answer
38 views

continuous extension and smooth extension of a function

Let $X$ be a metric space. Let $E$ be a subset of $X$. (1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such ...
0
votes
0answers
14 views

How do you detect when an iterated function converges / diverges and calculate limit accurate-enough?

See this post for a background on what I'm doing. So how many iterations $N$ of $P_c(z)$ does it take so that if $f(c) = P_c^N(z)$, and $g(c) = $ infinite iterations of $P_c(z)$, then $h \circ ...
1
vote
1answer
34 views

Problems with a series and a wrong problem statement suspected

I've been fighting with the following task: The series $\displaystyle \sum_{n=1}^\infty f_n(x)$ is convergent in the points $a$ and $b$, and the functions $f_n(x)$, $n \in \mathbb{N}$, are monotonic ...
2
votes
2answers
27 views

uniformly convergence on compact metric space

Let $K$ be a compact metric space. Let $\{f_n\}_{n=1}^\infty$ be a sequence of continuous functions on $K$ such that $f_n$ converges to a function $f$ pointwise on $K$. on Walt. Rudin's book ...
0
votes
1answer
42 views

Uniform convergence for a sequence of function

I have to show that if $f_n$ is a sequence of bounded functions that converges uniformly to $f$ on an interval I, then $f_n$ is uniformly bounded. I understand what uniformly bounded means, there ...
0
votes
0answers
24 views

ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...