0
votes
1answer
28 views

Composition of functions which is one-to-one.

$f:Y\rightarrow Z$ and $g:X\rightarrow Y$ If $f\circ g$ is one-to-one then which of the following must be true? 1.$g\circ f$ is one-to-one. 2.g is one-to-one. 3.f is one-to-one. 4.g is onto.
0
votes
3answers
43 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
1
vote
1answer
33 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
1
vote
0answers
32 views

Is there a name for this property of a real function?

Let $M=\sup_{x \in [0,1]^n} f(x)$ where $f:[0,1]^n \rightarrow \mathbb{R}$ is differentiable twice, and write $x=(x_1, \dots, x_n)$. Let $M_{x_i=0}=\sup_{x \in [0,1]^n:x_i=0} f(x)$ and ...
2
votes
3answers
112 views

Finding the maximum and minimum values of $f(x)=a^x+a^{1/x}$

Let $f(x)=a^x+a^{1/x}\ (x\gt 0)$ where $a\in\mathbb R$ is a constant. Question 1 : What is the maximum value of $f(x)$ for $0\lt a\lt 1$? Question 2 : What is the minimum value of $f(x)$ for ...
0
votes
2answers
30 views

Calculate random integer inside a range of real numbers

$$F : \Bbb R \times \Bbb R \rightarrow \Bbb N $$ $$F(\text{minReal},\ \text{maxReal}) = \text{randomInt} \in \left[\text{minReal},\ \text{maxReal}\right] $$ Let $r \in [0, 1)$ be a random value. How ...
1
vote
2answers
95 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
0answers
27 views

Functional characterization of zeroth law of thermodynamics [Sepration of Variables]

Zeroth law of thermodynamics is stated also as: If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C. This can be formulated ...
1
vote
1answer
19 views

How $n^d \times m([0, \frac{1}{n}[^d) = m([0, 1[^d)$ follows from translation invariance and (finite) additivity

In this StackExchange question (which itself seems to reference to an exercise in Terence Tao's lecture notes on introductory measure theory on his blog here), it's said that assuming "finite ...
1
vote
2answers
46 views

To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$

If $f: \mathbb R \to \mathbb R$ be a function such that for some $n_o \in \mathbb N$ , the $n_o$th iterate of $f$ has a unique fixed point $b$ , then how to prove that $f(b)=b$ ? I cant think of ...
0
votes
1answer
29 views

Existence of function satisfying given conditions?

Let $f:[0,1]\longrightarrow[0,1]$ be continuous, strictly increasing and $f(1)=1$. Suppose further that $f(x)>x$ for all $x\in[0,1)$. Is there any function satisfying the above conditions? My ...
1
vote
3answers
28 views

Search for two Real Valued functions.

Can we have two real valued functions $f_1$ and $f_2$ defined on $[a,b]$ such that $f_1(x)=f_2(x)$ for infinitely many points and $f_1(x)\neq f_2(x)$ for infinitely many points. ?
1
vote
2answers
63 views

Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
1
vote
1answer
25 views

Derive property from continuity - is this proof valid?

Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$. EDIT ...
12
votes
2answers
82 views

Function such that zeros$=$order of the derivative

Does there exist a function $f\in C^n(\mathbb{R},\mathbb{R})$ for $n\ge2$ such that $f^{(n)}$ has exactly $n$ zeros, $f^{(n-1)}$ has exactly $n-1$ zeros and so on ? Where $f^{(n)}$ is the nth ...
1
vote
2answers
47 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 ...
0
votes
3answers
61 views

If $f$ is continuous, so is $g=|f|$ [closed]

Prove that if $f$ is continuous, so is $g=|f|$. I need help on this. Thank you. Ok, this is my first time here. The definition of continuity i am using is that $f$ is continuous at $a$ if for any ...
0
votes
2answers
84 views

Prove that a function is bijective

So, the problem sounds like this. You have two bijective functions $f:\mathbb{N} \to A$, $g:\mathbb{N} \to B$. We define the function $ h:\mathbb{N} \to A \cup B $, defined as: $$ h(n) = ...
11
votes
1answer
281 views

Additive functional inequality

The function $f:R_+\to R_+$ is continuously differentiable and increasing. Also, $f(0)=0$ and $f(\infty)=\infty$. Continuity and differentiability of higher orders can be assumed if necessary. ...
2
votes
1answer
28 views

Question about the local maxima of a funciton

Assume $f(x_1,x_2,\dots,x_n)$ is a smooth, continuos, differentiable function, and let we want to check if $(x'_1,x'_2,\dots,x'_n)$ is the local maxima or not. Assume the first order condition is ...
1
vote
3answers
55 views

Show $f$ is uniformly continuous

Let $f$ continuous function on $[0,\infty)$. Lets assume there are $a,b$ such that: $\lim_{x\rightarrow \infty} f(x)-(ax+b) = 0$. Prove $f$ is uniformly continuous on $[0,\infty)$. Well, At ...
1
vote
4answers
32 views

Lowest possible value of a function with derivative greater than 2

I have the following two problems, and want to attempt to solve them with Mathematical rigour(which I don't yet possess): Suppose that $f$ is differentiable on $[1,4]$ and is such that $f(1) = 10$ ...
0
votes
4answers
42 views

What is $f$? Finding where a function converges pointwise?

I have a question. Let $f_n(x) = x^{4n} + \frac1{n^2}$. AS $n \to \infty$, $f_n$ converges pointwise to a function $f$ on $[0,1]$ What is $f$? Now if I am understanding correctly, couldn't ...
1
vote
1answer
81 views

Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
2
votes
0answers
40 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 $p \mapsto A_p$ ...
1
vote
1answer
41 views

Handy way to find the $x$ value where $\sin x \cos \left( \frac{\pi}{2} \sin x \right)$ is maximum?

Like in the title, is there a handy way to compute the $x$ values for which the function $$f(x) = \sin x \cos \left( \frac{\pi}{2} \sin x \right)$$ reaches its maxima? The derivative is $$f'(x) = ...
18
votes
2answers
278 views

Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
0
votes
0answers
24 views

Proof or disproof the existence of a continuous not strictly unimodal convex (strictly convex, not convex) function

I have to proof the existence or disproof the existence of a continuous function $f:[-1, 1]\longrightarrow \mathbb{R}$ with the following properties $f$ is not strictly unimodal $f$ is either: (a) ...
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 ...
3
votes
1answer
46 views

Behavior of Two Functions

If two functions can be shown to agree at an infinite number of points, what additional information would be required to show that these two functions are equivalent? For example, if two polynomials ...
16
votes
3answers
347 views

The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$. My attempt : f is $2\pi$ periodic and $\mathcal{C}^3$, we have : ...
0
votes
1answer
42 views

Composition of functions is constant.

If $f\circ g$ ($f$ composed with $g$) is constant, then which of the following is constant? $f$ $g$ $g \circ f$ ($g$ composed with $f$) Both $f$ and $g$ Both $g$ and $g \circ f$ ...
9
votes
5answers
930 views

Functions that are continuous only at two points?

I need to find a function $f:\mathbb{R}\to\mathbb{R}$ which is continuous only at two points, but discontinuous everywhere else. How on earth would I go about doing this? I can't think of any ...
4
votes
2answers
91 views

Show that there exist $g\in C^{\infty}$ such that $f(x)=g(x^2)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$, even,. Show that there exist $g\in C^{\infty}(\mathbb{R^+},\mathbb{R})$ such that for all $x\in \mathbb{R}, f(x)=g(x^2)$ My attempt : Let $g$ ...
5
votes
1answer
79 views

Show that $f$ is bounded at $(0,+\infty)$.

Show that a function $f\in C^{1}\bigl((0,+\infty)\bigr)$ which satisfy $$f'(x)=\frac{1}{1+x^{4}+\cos f(x)},\, x>0$$ is bounded at $(0,+\infty)$. My attempt : I would like to prove that ...
1
vote
0answers
26 views

Additive like function representation [duplicate]

Let's $f:R\rightarrow R,~r\gt0.$ And we have that for every $x,y\in R\Rightarrow |f(x+y)-f(x)-f(y)|\le r.$ Prove that there are $h:R \rightarrow R$ additive and $g:R \rightarrow [-r,r]$ functions such ...
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 ...
3
votes
1answer
114 views

Finding the maximum value of $\displaystyle \frac{x^n-a\sin x}{x^{n+1}}\ (x\gt 0)$

For $a\ge 1\in\mathbb R,n\ge 1\in\mathbb N$, let us define $f(x)$ as $$f(x)=\frac{x^n-a\sin x}{x^{n+1}}\ \ \ (x\gt 0).$$ Also, let $M(a,n)$ be the maximum value of $f(x).$ We may be able to know the ...
1
vote
1answer
22 views

conditions for Convergence of sequence of functions

Suppose $\{f_n\}$, $n \in \mathbb{N}$ is a sequence of a positive real-valued functions defined on $[0, T]$ and continuous on $(0, T)$. If {$f_n$} satisfies the following conditions : $f_n( iT/2^n ) ...
3
votes
1answer
25 views

Show for whih values this following function is continuous

For the function $f: [0,2 \pi] \rightarrow \mathbb{R}$ ,state at which points $c \in [0, \pi]$ is $f$ continuous or discontinuous. $$f(x)=\begin{array}{cc} ( & \begin{array}{cc} ...
2
votes
1answer
43 views

How to prove that $f_n(x) = \frac{1}{1+n^{2}x^{2}}$ is continuous on $[0,1]$?

I am having trouble verifying continuity. This seems like a very simple problem but I am not sure if my approach is correct: To prove that $f_n(x) = \frac{1}{1+n^{2}x^{2}}$ is continuous on $[0,1]$, ...
1
vote
1answer
88 views

Prove there are $b,c$ such that $f(x)=\frac{a}{2}x^2 + bx + c$

Let $f:I\rightarrow \mathbb{R}$, a differetiable twice. It is given that $\forall x\in I: f''(x) = a$. Prove there are $b,c$ such that $f(x)=\frac{a}{2}x^2 + bx + c$. I was guided to use LMVT ...
6
votes
6answers
199 views

Prove that $f(x)\equiv0$ on $\left[0,1\right]$

Let $f(x)$ differentiable on $\left[0,1\right]$ such that $f(0) = 0$. Also, assuming that $\forall x\in \left[0,1\right]:\left|f'(x)\right| \le \left|f(x)\right|$. Prove that $f(x)\equiv 0$ What I ...
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
106 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 ...
2
votes
4answers
97 views

Prove: for $\forall x\ne 0, \cos x < 1 - {x^2\over 2} + {x^4\over 24}$

Prove: for $\forall x\ne 0, \cos x < 1 - {x^2\over 2} + {x^4\over 24}$ What I did: We can prove: $${\cos x -1 + {x^2\over 2} \over {x^4\over 24}} < 1$$ Lets define: $f(x) = \cos x -1 + ...
0
votes
2answers
49 views

$f$ is strictly increasing and $g$ is decreasing. How to find whether $f \circ g$ and $g\circ f$ are increasing or decreasing?

$f$ is strictly increasing and $g$ is decreasing. How to find whether $f\circ g$ and $g\circ f$ are increasing, decreasing, strictly increasing or strictly decreasing? This is how I did, $f \circ ...
0
votes
2answers
44 views

$\cos(x)$ domain and range

I'd like to refer the following answer: http://math.stackexchange.com/a/628992/130682 @robjohn claims that: ...
0
votes
2answers
46 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
0answers
58 views

Continuous Function Uniformly Bounded

Assume $f : I → R$ is a non-decreasing on an open interval $I$ and that $f$ satisfies the Intermediate Value Theorem between any two points of $I$ (that is, for any $a < b$ in $I$ and any $L$ ...