Found problems: 35
Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations:
$$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$
Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as
$$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$
doesn't have limit at $ \infty . $
[i]Romeo Ilie[/i]
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$?
[i]Proposed by David Yang[/i]
Let be two real numbers $ a<b, $ a natural number $ n\ge 2, $ and a continuous function $ f:[a,b]\longrightarrow (0,\infty ) $ whose image contains $ 1 $ and that admits a primitive $ F:[a,b]\longrightarrow [a,b] . $ Prove that there is a real number $ c\in (a,b) $ such that
$$ (\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(b) -(\underbrace{F\circ\cdots\circ F}_{\text{n times}} )(a) =(f(c))^{n+1} (b-a) $$
[i]Vlad Mihaly[/i]
Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times.
Prove that the application $ \mathbb{R}\ni x\mapsto 2x+ \{ x\} $ and its inverse are bijective and continuous.
Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$,
\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]
a) Show that $I(f) < 3$, for any $f \in \mathcal{F}$.
b) Determine $\sup\{I(f) \mid f \in \mathcal{F}\}$.
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} ,f(x)=\left\{\begin{matrix} \sin x , & x\not\in\mathbb{Q} \\ 0, & x\in\mathbb{Q}\end{matrix}\right. . $
[b]a)[/b] Determine the maximum length of an interval $ I\subset\mathbb{R} $ such that $ f|_I $ is discontinuous everywhere, yet has the intermediate value property.
[b]b)[/b] Study the convergence of the sequence $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ defined by $ x_0\in (0,\pi /2),x_{n+1}=f\left( x_n\right),\forall n\ge 0. $
Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions with $g(x)=2f(x)+f(x^2),$ for all $x \in \mathbb{R}.$
a) Prove that, if $f$ is bounded in a neighbourhood of the origin and $g$ is continuous in the origin, then $f$ is continuous in the origin.
b) Provide an example of function $f$, discontinuous in the origin, for which the function $g$ is continuous in the origin.
Let $f:[a,b]\to[a,b]$ be a continuous function. It is known that there exist $\alpha,\beta\in (a,b)$ such that $f(\alpha)=a$ and $f(\beta)=b$. Prove that the function $f\circ f$ has at least three fixed points.
Let $f \colon [0,1] \to \mathbb{R}$ be a continuous function which is differentiable on $(0,1)$. Prove that either $f(x) = ax + b$ for all $x \in [0,1]$ for some constants $a,b \in \mathbb{R}$ or there exists $t \in (0,1)$ such that $|f(1) - f(0)| < |f'(t)|$.
Let $a, b, c \in \mathbb{R}$ be such that
$$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$
We say that a function $f$ is a [i]Palić function[/i] if $f: \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous and satisfies
$$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$
for all $x, y, z \in \mathbb{R}.$
Prove that any Palić function is infinitely many times differentiable and find all Palić functions.
Prove that if $f(x)$ is continuous for $a\leq x \leq b$ and
$$\int_{a}^{b} x^n f(x) \, dx =0$$
for $n=0,1,2, \ldots,$ then $f(x)$ is identically zero on $a \leq x \leq b.$
Let be a real function that admits finite right-limits everywhere. Prove that the function that maps every real number to its right-limit is right-continuous everywhere.
[i]Tolosi Marin[/i]
Let $a < b$ be real numbers and let $f : (a, b) \to \mathbb{R}$ be a function such that the functions $g : (a, b) \to \mathbb{R}$, $g(x) = (x - a) f(x)$ and $h : (a, b) \to \mathbb{R}$, $h(x) = (x - b) f(x)$ are increasing. Show that the function $f$ is continuous on $(a, b)$.
Determine all continuous functions $f:\mathbb{R}\to\mathbb{R}$ for which $f(1)=e$ and \[f(x+y)=e^{3xy}\cdot f(x)f(y),\]for all real numbers $x{}$ and $y{}$.
Discuss $ \lim_{x\to 0}\frac{\lambda +\sin\frac{1}{x} \pm\cos\frac{1}{x}}{x} . $
Let $f(x)=(x-1)(x-2)(x-3)$. Consider $g(x)=min\{f(x),f'(x)\}$. Then the number of points of discontinuity are
[list=1]
[*] 0
[*] 1
[*] 2
[*] More than 2
[/list]
Prove that, if a continuous function has limits at $ \pm\infty , $ and these are equal, then it touches its maximum or minimum at one point.
Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property:
$$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$
[b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal.
[b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $
A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $
[b]a)[/b] Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $
[b]b)[/b] Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a bounded function in some neighbourhood of $ 0, $ such that there are three real numbers $ a>0, b>1, c $ with the property that
$$ f(ax)=bf(x)+c,\quad\forall x\in\mathbb{R} . $$
Show that $ f $ is continuous at $ 0 $ if and only if $ c=0. $
Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$
Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $
[i]Radu Gologan[/i]
Give an example of a sequence of continuous functions on $\mathbb R$ converging pointwise to $0$ which is not uniformly convergent on any nonempty open set.
Let be areal number $ r, $ a nonconstant and continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with period $ T $ and $ F $ be its primitive having $ F(0)=0. $ Define the funtion $ g:\mathbb{R}\longrightarrow\mathbb{R} $ as
$$ g(x)=\left\{\begin{matrix} f(1/x), & x\neq 0 \\ r, & x=0 \end{matrix}\right. $$
Prove that:
[b]a)[/b] the image of $ f $ is closed.
[b]b)[/b] $ g $ has the intermediate value property if and only if $ r\in f\left(\mathbb{R}\right) . $
[b]c)[/b] $ g $ is primitivable if and only if $ r=\frac{F(T)}{T} . $