Found problems: 14
Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.
Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.
The function $f(x)$, defined on the entire real line, is known but that for any $a > 1 $ the function $f(x)+f(ax)$ is continuous on the entire line. Prove that $f(x)$ is also continuous along the entire line.
Justify if continuity can be affirmed, denied or cannot be decided in the point$ x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases .
a) It is known only that for all natural $n$: $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$.
b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$.
c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$.
Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.
The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$. Show that $|f(x)| \le 1$ for all $x$.
Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.
Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.
Let $f:\mathbb R\to\mathbb R$ be a continuous function. Do there exist continuous functions $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ such that $f(x)=g(x)\sin x+h(x)\cos x$ holds for every $x\in\mathbb R$?
Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$
The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous.
(A.I. Plotkin, Leningrad)
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that for all $x\in\mathbb{R}$ and for all $t\geqslant 0$, $$f(x)=f(e^tx)$$ Show that $f$ is a constant function.
A function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is said to be [i]continuous in each variable separately [/i] if, for each fixed value $y_0$ of $y$, the function $f(x, y_0)$ is contnuous in the usual sense as a function in $x,$ and similarly $f(x_0 , y)$ is continuous as a function of $y$ for each fixed $x_0$.
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous in each variable separately. Show that there exists a sequence of continuous functions $g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that
$$f(x,y) =\lim_{n\to \infty}g_{n}(x,y)$$
for all $(x,y)\in \mathbb{R}^{2}.$
Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.