0
votes
0answers
5 views

Roots Analysis. $f:[a,b]-->R$ continuous and for every x there is a y such as that$|f(y)|<|f(x)|/2$ .Show that exists ξ such that $ f(ξ)=0$

$f:[a,b]-->R$ continuous and for every x there is a y such as that $|f(y)|<|f(x)|/2$ .Show that exists ξ such that $f(ξ)=0$
0
votes
0answers
22 views

Continuity of function $f(x)=[x]\sinπx$

Examine the continuity of $f(x)=[x]\sinπx$ for $x \in\mathbb{R}$, where $[x]$ is the integral part of $x$.
0
votes
1answer
41 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
1
vote
0answers
20 views

Rules to evaluate Suprema and Infima

Note: All functions considered are supposed to be bounded. Nowhere I found rules to evaluate suprema and infima. Obviously, $$ c \cdot \sup_{x\in \mathbb R} f(x) = \sup_{x\in\mathbb R} cf(x) ...
-1
votes
2answers
35 views

Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
0
votes
2answers
71 views

Real and imaginary parts of a complex-valued function

How do you get a complex-valued function $ f(z) = f(x+iy) = \frac{z^{s-1}}{e^{-z}-1}, $ where $s$ is a constant complex number and $z$ is a complex variable, into the form: $ f(x+iy) = a(x,y) + ...
0
votes
1answer
18 views

Are all singular functions of bounded variation?

Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...
1
vote
1answer
26 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
2
votes
2answers
56 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
1
vote
0answers
15 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
-1
votes
1answer
29 views

An example of a twice continuously differentiable and bounded function. [on hold]

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
31 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 ...
2
votes
1answer
35 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
0
votes
0answers
50 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
52 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
47 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
127 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
40 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
107 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
52 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?
3
votes
1answer
35 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
84 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
46 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
72 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
2answers
66 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
43 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
33 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
30 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
43 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
24 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
25 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 ...