Found problems: 10
Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that satisfies the conditions:
$ \text{(i)}\quad \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x) $
$ \text{(ii)}\quad f $ has Darboux’s property
[b]a)[/b] Prove that the limits of $ f $ at $ \pm\infty $ exist.
[b]b)[/b] Is possible for the limits from [b]a)[/b] to be finite?
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 be a function $ f $ of class $ \mathcal{C}^1[a,b] $ whose derivative is positive. Prove that there exists a real number $ c\in (a,b) $ such that
$$ f(f(b))-f(f(a))=(f'(c))^2(b-a) . $$
Let $ I $ be an open real interval, and let be two functions $ f,g:I\longrightarrow\mathbb{R} $ satisfying the identity:
$$ x,y\in I\wedge x\neq y\implies\frac{f(x)-g(y)}{x-y} +|x-y|\ge 0. $$
[b]a)[/b] Prove that $ f,g $ are nondecreasing.
[b]b)[/b] Give a concrete example for $ f\neq g. $
Let $ f:(0,1)\longrightarrow \mathbb{R} $ be a bounded function having the property of Darboux. Then:
[b]a)[/b] There exists $ g:[0,1)\longrightarrow\mathbb{R} $ with Darboux’s property such that $ g\bigg|_{(0,1)} =f\bigg|_{(0,1)} . $
[b]b)[/b] The function above is uniquely determined if and only if $ f $ has limit at $ 0. $
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} . $
Find all functions, $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ that have the properties that $ f^2 $ is differentiable and $ f=\left( f^2 \right)' . $
Let $ f:[0,1]\longrightarrow (0,\infty ) $ be a continuous function and $ \left( b_n \right)_{n\ge 1} $ be a sequence of numbers from the interval $ (0,1) $ that converge to $ 0. $
[b]a)[/b] Demonstrate that for any fixed $ n, $ the equation $ F(x)=b_nF(1)+\left( 1-b_n\right) F(0) $ has an unique solution, namely $ x_n, $ where $ F $ is a primitive of $ f. $
[b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{x_n}{b_n} . $
Let be a real function that has the intermediate value property and is monotone on the irrationals. Show that it's continuous.
[i]Mihai Piticari[/i]
[b]a)[/b] Give an example of function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ that admits a primitive $ F:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ having the property that $ F^e $ is a primitive of $ f^e. $
[b]b)[/b] Prove that there is no derivable function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that has a primitive $ G:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ e^G $ is a primitive of $ e^g. $